Use substitution to express each of the following integrals as a multiple of $\int_{a}^{b}(1 / w) d w$ for some $a$ and $b$ Then evaluate the integrals.

(a) $\int_{0}^{1} \frac{x}{1+x^{2}} d x$

(b) $\int_{0}^{\pi / 4} \frac{\sin x}{\cos x} d x$

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(a) Find the derivatives of $\sin \left(x^{2}+1\right)$ and $\sin \left(x^{3}+1\right)$

(b) Use your answer to part (a) to find antiderivatives of:

(i) $x \cos \left(x^{2}+1\right)$

(ii) $x^{2} \cos \left(x^{3}+1\right)$

(c) Find the general antiderivatives of:

(i) $x \sin \left(x^{2}+1\right)$

(ii) $x^{2} \sin \left(x^{3}+1\right)$

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Find the integrals. Check your answers by differentiation.

$$\int e^{3 x} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int t e^{t^{2}} d t$$

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Find the integrals. Check your answers by differentiation.

$$\int e^{-x} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int 25 e^{-0.2 t} d t$$

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Find the integrals. Check your answers by differentiation.

$$\int \sin (2 x) d x$$

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Find the integrals. Check your answers by differentiation.

$$\int t \cos \left(t^{2}\right) d t$$

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Find the integrals. Check your answers by differentiation.

$$\int \sin (3-t) d t$$

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Find the integrals. Check your answers by differentiation.

$$\int x e^{-x^{2}} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int(r+1)^{3} d r$$

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Find the integrals. Check your answers by differentiation.

$$\int y\left(y^{2}+5\right)^{8} d y$$

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Find the integrals. Check your answers by differentiation.

$$\int x^{2}\left(1+2 x^{3}\right)^{2} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int t^{2}\left(t^{3}-3\right)^{10} d t$$

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Find the integrals. Check your answers by differentiation.

$$\int x\left(x^{2}+3\right)^{2} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int x\left(x^{2}-4\right)^{7 / 2} d x$$

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Find the integrals. Check your answers by differentiation.

$$-\int y^{2}(1+y)^{2} d y$$

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Find the integrals. Check your answers by differentiation.

$$\int(2 t-7)^{73} d t$$

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Find the integrals. Check your answers by differentiation.

$$\int x^{2} e^{x^{3}+1} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{d y}{y+5}$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{1}{\sqrt{4-x}} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int\left(x^{2}+3\right)^{2} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \sin \theta(\cos \theta+5)^{7} d \theta$$

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Find the integrals. Check your answers by differentiation.

$$\int \sqrt{\cos 3 t} \sin 3 t \, d t$$

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Find the integrals. Check your answers by differentiation.

$$\int \sin ^{6} \theta \cos \theta d \theta$$

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Find the integrals. Check your answers by differentiation.

$$\int \sin ^{3} \alpha \cos \alpha d \alpha$$

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Find the integrals. Check your answers by differentiation.

$$\int \sin ^{6}(5 \theta) \cos (5 \theta) d \theta$$

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Find the integrals. Check your answers by differentiation.

$$\int \tan (2 x) d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{(\ln z)^{2}}{z} d z$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{e^{t}+1}{e^{t}+t} d t$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{(t+1)^{2}}{t^{2}} d t$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{y}{y^{2}+4} d y$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{d x}{1+2 x^{2}}$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{d x}{\sqrt{1-4 x^{2}}}$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{e^{\sqrt{y}}}{\sqrt{y}} d y$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{1+e^{x}}{\sqrt{x+e^{x}}} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{e^{x}}{2+e^{x}} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{x+1}{x^{2}+2 x+19} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{t}{1+3 t^{2}} d t$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \frac{x \cos \left(x^{2}\right)}{\sqrt{\sin \left(x^{2}\right)}} d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \sinh 3 t \, d t$$

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Find the integrals. Check your answers by differentiation.

$$\int \cosh x \, d x$$

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Find the integrals. Check your answers by differentiation.

$$\int \cosh (2 w+1) d w$$

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Find the integrals. Check your answers by differentiation.

$$\int(\sinh z) e^{\cosh z} d z$$

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Find the integrals. Check your answers by differentiation.

$$\int \cosh ^{2} x \sinh x d x$$

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Find the integrals. Check your answers by differentiation.

$$\int x \cosh x^{2} d x$$

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Find the general antiderivative. Check your answers by differentiation.

$$p(t)=\pi t^{3}+4 t$$

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Find the general antiderivative. Check your answers by differentiation.

$$f(x)=\sin 3 x$$

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Find the general antiderivative. Check your answers by differentiation.

$$f(x)=2 x \cos \left(x^{2}\right)$$

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Find the general antiderivative. Check your answers by differentiation.

$$r(t)=12 t^{2} \cos \left(t^{3}\right)$$

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Find the general antiderivative. Check your answers by differentiation.

$$f(x)=\sin (2-5 x)$$

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Find the general antiderivative. Check your answers by differentiation.

$$f(x)=e^{\sin x} \cos x$$

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Find the general antiderivative. Check your answers by differentiation.

$$f(x)=\frac{x}{x^{2}+1}$$

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Find the general antiderivative. Check your answers by differentiation.

$$f(x)=\frac{1}{3 \cos ^{2}(2 x)}$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{0}^{\pi} \cos (x+\pi) d x$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{0}^{1 / 2} \cos (\pi x) d x$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{0}^{\pi / 2} e^{-\cos \theta} \sin \theta d \theta$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{1}^{2} 2 x e^{x^{2}} d x$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{-1}^{e-2} \frac{1}{t+2} d t$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{1}^{4} \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

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Use the Fundamental Theorem to calculate the definite integrals.

$$\int_{0}^{2} \frac{x}{\left(1+x^{2}\right)^{2}} d x$$

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Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.

$$\int_{-1}^{3}\left(x^{3}+5 x\right) d x$$

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Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.

$$\int_{-1}^{1} \frac{1}{1+y^{2}} d y$$

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Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.

$$\int_{1}^{3} \frac{1}{x} d x$$

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Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.

$$\int_{1}^{3} \frac{d t}{(t+7)^{2}}$$

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Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.

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Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.

$$\int_{1}^{2} \frac{\sin t}{t} d t$$

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Show the two integrals are equal using a substitution.

$$\int_{0}^{\pi / 3} 3 \sin ^{2}(3 x) d x=\int_{0}^{\pi} \sin ^{2}(y) d y$$

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Show the two integrals are equal using a substitution.

$$\int_{1}^{2} 2 \ln \left(s^{2}+1\right) d s=\int_{1}^{4} \frac{\ln (t+1)}{\sqrt{t}} d t$$

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Show the two integrals are equal using a substitution.

$$\int_{1}^{e}(\ln w)^{3} d w=\int_{0}^{1} z^{3} e^{z} d z$$

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Show the two integrals are equal using a substitution.

$$\int_{0}^{\pi} x \cos (\pi-x) d x=\int_{0}^{\pi}(\pi-t) \cos t d t$$

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Using the substitution $w=x^{2},$ find a function $g(w)$ such that $\int_{\sqrt{a}}^{\sqrt{b}} d x=\int_{a}^{b} g(w) d w$ for all $0<a<b$.

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Using the substitution $w=e^{x},$ find a function $g(w)$ such that $\int_{a}^{b} e^{-x} d x=\int_{e^{a}}^{e^{b}} g(w) d w$ for all $a<b$.

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Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.

$$\int \frac{e^{x} d x}{1+e^{2 x}} \text { and } \int \frac{\cos x d x}{1+\sin ^{2} x}$$

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Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.

$$\int \frac{\ln x}{x} d x \text { and } \, \int x \, d x$$

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Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.

$$\int e^{\sin x} \cos x d x \quad \text { and } \quad \int \frac{e^{\arcsin x}}{\sqrt{1-x^{2}}} d x$$

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Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.

$$\int(\sin x)^{3} \cos x d x \quad \text { and } \quad \int\left(x^{3}+1\right)^{3} x^{2} d x$$

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Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.

$$\int \sqrt{x+1} d x \text { and } \, \int \frac{\sqrt{1+\sqrt{x}}}{\sqrt{x}} d x$$

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Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:

$$\begin{array}{c|c|c|r|r|r}

\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\

\hline f(x) & 5 & 7 & 8 & 10 & 11 \\

\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\

\hline

\end{array}$$

$$\int_{0}^{1} f^{\prime}(x) \sin f(x) d x$$

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Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:

$$\begin{array}{c|c|c|r|r|r}

\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\

\hline f(x) & 5 & 7 & 8 & 10 & 11 \\

\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\

\hline

\end{array}$$

$$\int_{1}^{3} f^{\prime}(x) e^{f(x)} d x$$

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Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:

$$\begin{array}{c|c|c|r|r|r}

\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\

\hline f(x) & 5 & 7 & 8 & 10 & 11 \\

\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\

\hline

\end{array}$$

$$\int_{1}^{3} \frac{f^{\prime}(x)}{f(x)} d x$$

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Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:

$$\begin{array}{c|c|c|r|r|r}

\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\

\hline f(x) & 5 & 7 & 8 & 10 & 11 \\

\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\

\hline

\end{array}$$

$$\int_{0}^{1} e^{x} f^{\prime}\left(e^{x}\right) d x$$

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Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:

$$\begin{array}{c|c|c|r|r|r}

\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\

\hline f(x) & 5 & 7 & 8 & 10 & 11 \\

\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\

\hline

\end{array}$$

$$\int_{1}^{e} \frac{f^{\prime}(\ln x)}{x} d x$$

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Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:

$$\begin{array}{c|c|c|r|r|r}

\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\

\hline f(x) & 5 & 7 & 8 & 10 & 11 \\

\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\

\hline

\end{array}$$

$$\int_{0}^{1} f^{\prime}(x)(f(x))^{2} d x$$

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Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:

$$\begin{array}{c|c|c|r|r|r}

\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\

\hline f(x) & 5 & 7 & 8 & 10 & 11 \\

\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\

\hline

\end{array}$$

$$\int_{0}^{\pi / 2} \sin x \cdot f^{\prime}(\cos x) d x$$

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Find an expression for the integral which contains $g$ but no integral sign.

$$\int g^{\prime}(x)(g(x))^{4} d x$$

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Find an expression for the integral which contains $g$ but no integral sign.

$$\int g^{\prime}(x) e^{g(x)} d x$$

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Find an expression for the integral which contains $g$ but no integral sign.

$$\int g^{\prime}(x) \sin g(x) d x$$

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Find an expression for the integral which contains $g$ but no integral sign.

$$\int g^{\prime}(x) \sqrt{1+g(x)} d x$$

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Find a substitution $w$ and constants $k, n$ so that the integral has the form $\int k w^{n} d w$.

$$\int x^{2} \sqrt{1-4 x^{3}} d x$$

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Find a substitution $w$ and constants $k, n$ so that the integral has the form $\int k w^{n} d w$.

$$\int \frac{\cos t}{\sin t} d t$$

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Find a substitution $w$ and constants $k, n$ so that the integral has the form $\int k w^{n} d w$.

$$\int \frac{2 x d x}{\left(x^{2}-3\right)^{2}}$$

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Find constants $k, n, w_{0}, w_{1}$ so the the integral has the form $\int_{w_{0}}^{w_{1}} k w^{n} d w$.

$$\int_{1}^{5} \frac{3 x d x}{\sqrt{5 x^{2}+7}}, \quad w=5 x^{2}+7$$

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Find constants $k, n, w_{0}, w_{1}$ so the the integral has the form $\int_{w_{0}}^{w_{1}} k w^{n} d w$.

$$\int_{0}^{5} \frac{2^{x} d x}{2^{x}+3}, \quad w=2^{x}+3$$

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Find constants $k, n, w_{0}, w_{1}$ so the the integral has the form $\int_{w_{0}}^{w_{1}} k w^{n} d w$.

$$\int_{\pi / 12}^{\pi / 4} \sin ^{7}(2 x) \cos (2 x) d x, \quad w=\sin 2 x$$

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Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.

$$\int x e^{-x^{2}} d x$$

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Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.

$$\int e^{\sin \phi} \cos \phi d \phi$$

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Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.

$$\int \sqrt{e^{r}} d r$$

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Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.

$$\int \frac{z^{2} d z}{e^{-z^{3}}}$$

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Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.

$$\int e^{2 t} e^{3 t-4} d t$$

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In Problems $112-113,$ find a substitution $w$ and constants $a, b, A$ so that the integral has the form $\int_{a}^{b} A e^{w} d w$

$$\int_{3}^{7} e^{2 t-3} d t$$

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Find a substitution $w$ and constants $a, b, A$ so that the integral has the form $\int_{a}^{b} A e^{w} d w$.

$$\int_{0}^{1} e^{\cos (\pi t)} \sin (\pi t) d t$$

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Integrate:

(a) $\int \frac{1}{\sqrt{x}} d x$

(b) $\int \frac{1}{\sqrt{x+1}} d x$

(c) $\int \frac{1}{\sqrt{x}+1} d x$

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If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate.

(a) $\int x \sin \left(x^{2}\right) d x$

(b) $\int x^{2} \sin x d x$

(c) $\int \frac{x^{2}}{1+x^{2}} d x$

(d) $\int \frac{x}{\left(1+x^{2}\right)^{2}} d x$

(e) $\int x^{3} e^{x^{2}} d x$

(f) $\int \frac{\sin x}{2+\cos x} d x$

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Find the exact area.

Under $f(x)=\sinh (x / 2)$ between $x=0$ and $x=2$.

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Find the exact area.

$$\text { Under } f(\theta)=\left(e^{\theta+1}\right)^{3} \text { for } 0 \leq \theta \leq 2$$

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Find the exact area.

Between $e^{t}$ and $e^{t+1}$ for $0 \leq t \leq 2$.

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Find the exact area.

Under one arch of the curve $V(t)=V_{0} \sin (\omega t),$ where $V_{0}>0$ and $\omega>0$.

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Find the exact average value of $f(x)=1 /(x+1)$ on the interval $x=0$ to $x=2 .$ Sketch a graph showing the function and the average value.

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Let $g(x)=f(2 x) .$ Show that the average value of $f$ on the interval $[0,2 b]$ is the same as the average value of $g$ on the interval $[0, b]$.

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Suppose $\int_{0}^{2} g(t) d t=5 .$ Calculate the following:

(b) $\int_{0}^{2} g(2-t) d t$

(a) $\int_{0}^{4} g(t / 2) d t$

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Suppose $\int_{0}^{1} f(t) d t=3 .$ Calculate the following:

(a) $\int_{0}^{0.5} f(2 t) d t$

(b) $\int_{0}^{1} f(1-t) d t$

(c) $\int_{1}^{1.5} f(3-2 t) d t$

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(a) Calculate exactly: $\int_{-\pi}^{\pi} \cos ^{2} \theta \sin \theta d \theta$.

(b) Calculate the exact area under the curve $y=\cos ^{2} \theta \sin \theta$ between $\theta=0$ and $\theta=\pi$.

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Find $\int 4 x\left(x^{2}+1\right) d x$ using two methods:

(a) Do the multiplication first, and then antidifferentiate.

(b) Use the substitution $w=x^{2}+1$.

(c) Explain how the expressions from parts (a) and (b) are different. Are they both correct?

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(a) Find $\int \sin \theta \cos \theta d \theta$.

(b) You probably solved part (a) by making the substitution $w=\sin \theta$ or $w=\cos \theta .$ (If not, go back and do it that way.) Now find $\int \sin \theta \cos \theta d \theta$ by making the other substitution.

(c) There is yet another way of finding this integral which involves the trigonometric identities $\sin (2 \theta)=2 \sin \theta \cos \theta$ $\cos (2 \theta)=\cos ^{2} \theta-\sin ^{2} \theta$.

Find $\int \sin \theta \cos \theta d \theta$ using one of these identities and then the substitution $w=2 \theta$.

(d) You should now have three different expressions for the indefinite integral $\int \sin \theta \cos \theta d \theta .$ Are they really different? Are they all correct? Explain.

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Find a substitution $w$ and constants $a, b, k$ so that the integral has the form $\int_{a}^{b} k f(w) d w$.

$$\int_{1}^{9} f(6 x \sqrt{x}) \sqrt{x} d x$$

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Find a substitution $w$ and constants $a, b, k$ so that the integral has the form $\int_{a}^{b} k f(w) d w$.

$$\int_{2}^{5} \frac{f\left(\ln \left(x^{2}+1\right)\right) x d x}{x^{2}+1}$$

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Find the solution of the initial value problem

$$

y^{\prime}=\tan x+1, \quad y(0)=1

$$

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Let $I_{m, n}=\int_{0}^{1} x^{m}(1-x)^{n} d x$ for constant $m, n .$ Show that $I_{m, n}=I_{n, m}$.

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Let $f(t)$ be the velocity in meters/second of a car at time $t$ in seconds. Give an integral for the change of position of the car

(a) For the time interval $0 \leq t \leq 60$.

(b) In terms of $T$ in minutes, for the same time interval.

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Over the past fifty years the carbon dioxide level in the atmosphere has increased. Carbon dioxide is believed to drive temperature, so predictions of future carbon dioxide levels are important. If $C(t)$ is carbon dioxide level in parts per million (ppm) and $t$ is time in years since 1950 three possible models are: $^{1}$

I $C^{\prime}(t)=1.3$

II $C^{\prime}(t)=0.5+0.03 t$

III $C^{\prime}(t)=0.5 e^{0.02 t}$

(a) Given that the carbon dioxide level was 311 ppm in $1950,$ find $C(t)$ for each model.

(b) Find the carbon dioxide level in 2020 predicted by each model.

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Let $f(t)$ be the rate of flow, in cubic meters per hour, of a flooding river at time $t$ in hours. Give an integral for the total flow of the river

(a) Over the 3 -day period $0 \leq t \leq 72$.

(b) In terms of time $T$ in days over the same 3 -day period.

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With $t$ in years since $2000,$ the population, $P,$ of the world in billions can be modeled by $P=6.1 e^{0.012 t}$

(a) What does this model predict for the world population in $2010 ?$ In $2020 ?$

(b) Use the Fundamental Theorem to predict the average population of the world between 2000 and 2010

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Oil is leaking out of a ruptured tanker at the rate of $r(t)=50 e^{-0.02 t}$ thousand liters per minute.

(a) At what rate, in liters per minute, is oil leaking out at $t=0 ?$ At $t=60 ?$

(b) How many liters leak out during the first hour?

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Throughout much of the $20^{\text {th }}$ century, the yearly consumption of electricity in the US increased exponentially at a continuous rate of $7 \%$ per year. Assume this trend continues and that the electrical energy consumed in 1900 was 1.4 million megawatt-hours.

(a) Write an expression for yearly electricity consumption as a function of time, $t,$ in years since 1900

(b) Find the average yearly electrical consumption throughout the $20^{\text {th }}$ century.

(c) During what year was electrical consumption closest to the average for the century?

(d) Without doing the calculation for part (c), how could you have predicted which half of the century the answer would be in?

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An electric current, $I(t),$ flowing out of a capacitor, decays according to $I(t)=I_{0} e^{-t},$ where $t$ is time. Find the charge, $Q(t),$ remaining in the capacitor at time $t$ The initial charge is $Q_{0}$ and $Q(t)$ is related to $I(t)$ by

$$

Q^{\prime}(t)=-I(t)

$$

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If we assume that wind resistance is proportional to velocity, then the downward velocity, $v,$ of a body of mass $m$ falling vertically is given by

$$

v=\frac{m g}{k}\left(1-e^{-k t / m}\right)

$$,

where $g$ is the acceleration due to gravity and $k$ is a constant. Find the height, $h$, above the surface of the earth as a function of time. Assume the body starts at height $h_{0}$.

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If we assume that wind resistance is proportional to the square of velocity, then the downward velocity, $v,$ of a falling body is given by

$$

v=\sqrt{\frac{g}{k}}\left(\frac{e^{t \sqrt{g k}}-e^{-t \sqrt{g k}}}{e^{t \sqrt{g k}}+e^{-t \sqrt{g k}}}\right)

$$

Use the substitution $w=e^{t \sqrt{g k}}+e^{-t \sqrt{g k}}$ to find the height, $h,$ of the body above the surface of the earth as a function of time. Assume the body starts at a height $h_{0}$.

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(a) Between 2000 and 2010 , ACME Widgets sold widgets at a continuous rate of $R=R_{0} e^{0.125 t}$ widgets per year, where $t$ is time in years since January 1 2000. Suppose they were selling widgets at a rate of 1000 per year on January $1,2000 .$ How many widgets did they sell between 2000 and $2010 ?$ How many did they sell if the rate on January 1,2000 was 1,000,000 widgets per year?

(b) In the first case ( 1000 widgets per year on January 1, 2000 ), how long did it take for half the widgets in the ten-year period to be sold? In the second case $(1,000,000 \text { widgets per year on January } 1,2000)$ when had half the widgets in the ten-year period been sold?

(c) In $2010,$ ACME advertised that half the widgets it had sold in the previous ten years were still in use. Based on your answer to part (b), how long must a widget last in order to justify this claim?

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The rate at which water is flowing into a tank is $r(t)$ gallons/minute, with $t$ in minutes.

(a) Write an expression approximating the amount of water entering the tank during the interval from time $t$ to time $t+\Delta t,$ where $\Delta t$ is small.

(b) Write a Riemann sum approximating the total amount of water entering the tank between $t=0$ and $t=5 .$ Write an exact expression for this amount.

(c) By how much has the amount of water in the tank changed between $t=0$ and $t=5$ if $r(t)=$ $20 e^{0.02 t} ?$

(d) If $r(t)$ is as in part (c), and if the tank contains 3000 gallons initially, find a formula for $Q(t),$ the amount of water in the tank at time $t$

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Explain what is wrong with the statement.

$$\int(f(x))^{2} d x=(f(x))^{3} / 3+C$$

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Explain what is wrong with the statement.

$$\int \cos \left(x^{2}\right) d x=\sin \left(x^{2}\right) /(2 x)+C$$

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Explain what is wrong with the statement.

$$\int_{0}^{\pi / 2} \cos (3 x) d x=(1 / 3) \int_{0}^{\pi / 2} \cos w d w$$

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Give an example of:

A possible $f(\theta)$ so that the following integral can be integrated by substitution:

$$

\int f(\theta) e^{\cos \theta} d \theta

$$

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Give an example of:

An indefinite integral involving $\sin \left(x^{3}-3 x\right)$ that can be evaluated by substitution.

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Decide whether the statements are true or false. Give an explanation for your answer.

$$\int f^{\prime}(x) \cos (f(x)) d x=\sin (f(x))+C$$

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Decide whether the statements are true or false. Give an explanation for your answer.

$$\int(1 / f(x)) d x=\ln |f(x)|+C$$

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Decide whether the statements are true or false. Give an explanation for your answer.

$\int t \sin \left(5-t^{2}\right) d t$ can be evaluated using substitution.

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