Section 1
The Indefinite Integral
Evaluate the given indefinite integral.$$\int 3 d x$$
Evaluate the given indefinite integral.$$\int\left(\pi^{2}-1\right) d x$$
Evaluate the given indefinite integral.
Evaluate the given indefinite integral.$$\int 5 x^{1 / 4} d x$$
Evaluate the given indefinite integral.$$\int \frac{1}{\sqrt[3]{x}} d x$$
Evaluate the given indefinite integral.$$\int \sqrt[3]{x^{2}} d x$$
Evaluate the given indefinite integral.$$\int\left(1-t^{-0.52}\right) d t$$
Evaluate the given indefinite integral.$$\int 10 w \sqrt{w} d w$$
Evaluate the given indefinite integral.$$\int\left(3 x^{2}+2 x-1\right) d x$$
Evaluate the given indefinite integral.$$\int\left(2 \sqrt{t}-t-\frac{9}{t^{2}}\right) d t$$
Evaluate the given indefinite integral.$$\int \sqrt{x}\left(x^{2}-2\right) d x$$
Evaluate the given indefinite integral.$$\int\left(\frac{5}{\sqrt[3]{s^{2}}}+\frac{2}{\sqrt{s^{3}}}\right) d s$$
Evaluate the given indefinite integral.$$\int(4 x+1)^{2} d x$$
Evaluate the given indefinite integral.$$\int(\sqrt{x}-1)^{2} d x$$
Evaluate the given indefinite integral.$$\int(4 w-1)^{3} d w$$
Evaluate the given indefinite integral.$$\int \frac{r^{2}-10 r+4}{r^{3}} d r$$
Evaluate the given indefinite integral.$$\int \frac{(x+1)^{2}}{\sqrt{x}} d x$$
Evaluate the given indefinite integral.$$\int \frac{x^{-1}-x^{-2}+x^{-3}}{x^{2}} d x$$
Evaluate the given indefinite integral.$$\int \frac{t^{3}-8 t+1}{(2 t)^{4}} d t$$
Evaluate the given indefinite integral.$$\int\left(4 \sin x-1+8 x^{-5}\right) d x$$
Evaluate the given indefinite integral.$$\int\left(-3 \cos x+4 \sec ^{2} x\right) d x$$
Evaluate the given indefinite integral.$$\int \frac{\sin t}{\cos ^{2} t} d t$$
Evaluate the given indefinite integral.$$\int \frac{2+3 \sin ^{2} x}{\sin ^{2} x} d x$$
Evaluate the given indefinite integral.$$\int\left(40-\frac{2}{\sec \theta}\right) d \theta$$
Evaluate the given indefinite integral.$$\int\left(8 x+1-9 e^{x}\right) d x$$
Evaluate the given indefinite integral.$$\int\left(15 x^{-1}-4 \sinh x\right) d x$$
Evaluate the given indefinite integral.$$\int \frac{2 x^{3}-x^{2}+2 x+4}{1+x^{2}} d x$$
Evaluate the given indefinite integral.$$\int \frac{x^{6}}{1+x^{2}} d x$$
Use a trigonometric identity to evaluate the given indefinite integral.$$\int \tan ^{2} x d x$$
Use a trigonometric identity to evaluate the given indefinite integral.$$\int \cos ^{2} \frac{x}{2} d x$$
Verify the given integration result by differentiation and the Chain Rule.$$\int \frac{1}{\sqrt{2 x+1}} d x=\sqrt{2 x+1}+C$$
Verify the given integration result by differentiation and the Chain Rule.$$\int\left(2 x^{2}-4 x\right)^{9}(x-1) d x=\frac{1}{40}\left(2 x^{2}-4 x\right)^{10}+C$$
Verify the given integration result by differentiation and the Chain Rule.$$\int \cos 4 x d x=\frac{1}{4} \sin 4 x+C$$
Verify the given integration result by differentiation and the Chain Rule.$$\int \sin x \cos x d x=\frac{1}{2} \sin ^{2} x+C$$
Verify the given integration result by differentiation and the Chain Rule.$$\int x \sin x^{2} d x=-\frac{1}{2} \cos x^{2}+C$$
Verify the given integration result by differentiation and the Chain Rule.$$\int \ln x d x=x \ln x-x+C$$
Verify the given integration result by differentiation and the Chain Rule.$$\int x e^{x} d x=x e^{x}-e^{x}+C$$
Perform the indicated operations.$$\frac{d}{d x} \int\left(x^{2}-4 x+5\right) d x$$
Perform the indicated operations.$$\int \frac{d}{d x}\left(x^{2}-4 x+5\right) d x$$
Solve the given differential equation.$$\frac{d y}{d x}=6 x^{2}+9$$
Solve the given differential equation.$$\frac{d y}{d x}=10 x+3 \sqrt{x}$$
Solve the given differential equation.$$\frac{d y}{d x}=\frac{1}{x^{2}}$$
Solve the given differential equation.$$\frac{d y}{d x}=\frac{(2+x)^{2}}{x^{5}}$$
Solve the given differential equation.$$\frac{d y}{d x}=1-2 x+\sin x$$
Solve the given differential equation.$$\frac{d y}{d x}=\frac{1}{\cos ^{2} x}$$
Find a function $y=f(x)$ whose graph passes through the point (2,3) that also satisfies the differential equation $d y / d x=2 x-1$.
Find a function $y=f(x)$ so that $d y / d x=1 / \sqrt{x}$ and $f(9)=1$.
Find a function $f$ such that $f^{\prime \prime}(x)=6, f^{\prime}(-1)=2,$ and $f(-1)=0$
Find a function $f$ such that $f^{\prime \prime}(x)=12 x^{2}+2$ for which the slope of the tangent line to its graph at (1,1) is $3 .$
If $f^{(n)}(x)=0,$ what is $f ?$.
The graph of a function $f$ is shown in blue. Of the graphs of functions $F, G,$ and $H$ whose graphs are shown in black, green, and red, respectively, which function is the graph of an antiderivative of $f ?$ State your reasoning.
A bucket that contains liquid is rotating about a vertical axis at a constant angular velocity $\omega .$ The shape of the cross-section of the rotating liquid in the $x y$ -plane is determined from$$\frac{d y}{d x}=\frac{\omega^{2}}{g} x$$With coordinate axes as shown in FIGURE $5.1 .5,$ find $y=f(x)$.
The ends of a beam of length $L$ rest on two supports as shown in FIGURE 5.1 .6 . With a uniform load on the beam, its shape (or elastic curve) is determined from$$E I y^{\prime \prime}=\frac{1}{2} q L x-\frac{1}{2} q x^{2}$$where $E, I,$ and $q$ are constants. Find $y=f(x)$ if $f(0)=0$ and $f^{\prime}(L / 2)=0 .$
Determine $f$.$$\int f(x) d x=\ln |\ln x|+C$$
Determine $f$.$$\int f(x) d x=x^{2} e^{x}-2 x e^{x}+2 e^{x}+C$$
Find a function $f$ such that $f^{\prime}(x)=x^{2}$ and $y=4 x+7$ is a tangent line to the graph of $f$.
Simplify the expression $e^{4 \int d x / x}$ as much as possible.
Determine which of the following two results is correct:$$\int(x+1)^{3} d x=\frac{1}{4}(x+1)^{4}+C$$or$$\int(x+1)^{3} d x=\frac{1}{4} x^{4}+x^{3}+\frac{3}{2} x^{2}+x+C ?$$
Given that $\frac{d}{d x} \sin \pi x=\pi \cos \pi x .$ Find an antiderivative $F$ of $\cos \pi x$ that has the property that $F\left(\frac{3}{2}\right)=0$.