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Section 1
Review of Formulas and Techniques
Evaluate the integral.$$\int \sin 6 x \, d x$$
Evaluate the integral.$$\int 3 \cos 4 x \, d x$$
Evaluate the integral.$$\int \sec 2 x \tan 2 x \, d x$$
Evaluate the integral.$$\int x \sec x^{2} \tan x^{2} d x$$
Evaluate the integral.$$\int e^{3-2 x} d x$$
Evaluate the integral.$$\int \frac{3}{e^{6 x}} d x$$
Evaluate the integral.$$\int \frac{4}{x^{1 / 3}\left(1+x^{2 / 3}\right)} d x$$
Evaluate the integral.$$\int \frac{2}{x^{1 / 4}+x} d x$$
Evaluate the integral.$$\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$$
Evaluate the integral.$$\int \frac{\cos (1 / x)}{x^{2}} d x$$
Evaluate the integral.$$\int_{0}^{\pi} \cos x e^{\sin x} d x$$
Evaluate the integral.$$\int_{0}^{\pi / 4} \sec ^{2} x e^{\tan x} d x$$
Evaluate the integral.$$\int_{-\pi / 4}^{0} \frac{\sin x}{\cos ^{2} x} d x$$
Evaluate the integral.$$\int_{\pi / 4}^{\pi / 2} \frac{1}{\sin ^{2} x} d x$$
Evaluate the integral.$$\int \frac{3}{16+x^{2}} d x$$
Evaluate the integral.$$\int \frac{2}{4+4 x^{2}} d x$$
Evaluate the integral.$$\int \frac{x^{2}}{1+x^{6}} d x$$
Evaluate the integral.$$\int \frac{x^{5}}{1+x^{6}} d x$$
Evaluate the integral.$$\int \frac{1}{\sqrt{4-x^{2}}} d x$$
Evaluate the integral.$$\int \frac{e^{x}}{\sqrt{1-e^{2 x}}} d x$$
Evaluate the integral.$$\int \frac{x}{\sqrt{1-x^{4}}} d x$$
Evaluate the integral.$$\int \frac{2 x^{3}}{\sqrt{1-x^{4}}} d x$$
Evaluate the integral.$$\int \frac{4}{5+2 x+x^{2}} d x$$
Evaluate the integral.$$\int \frac{4 x+4}{5+2 x+x^{2}} d x$$
Evaluate the integral.$$\int \frac{4 x}{5+2 x+x^{2}} d x$$
Evaluate the integral.$$\int \frac{x+1}{x^{2}+2 x+4} d x$$
Evaluate the integral.$$\int\left(x^{2}+4\right)^{2} d x$$
Evaluate the integral.$$\int x\left(x^{2}+4\right)^{2} d x$$
Evaluate the integral.$$\int \frac{1}{\sqrt{3-2 x-x^{2}}} d x$$
Evaluate the integral.$$\int \frac{x+1}{\sqrt{3-2 x-x^{2}}} d x$$
Evaluate the integral.$$\int \frac{1+x}{1+x^{2}} d x$$
Evaluate the integral.$$\int \frac{1}{\sqrt{x}+x} d x$$
Evaluate the integral.$$\int_{-2}^{-1} e^{\ln \left(x^{2}+1\right)} d x$$
Evaluate the integral.$$\int_{1}^{3} e^{2 \ln x} d x$$
Evaluate the integral.$$\int_{3}^{4} x \sqrt{x-3} d x$$
Evaluate the integral.$$\int_{0}^{1} x(x-3)^{2} d x$$
Evaluate the integral.$$\int_{0}^{2} \frac{e^{x}}{1+e^{2 x}} d x$$
Evaluate the integral.$$\int_{-1}^{0} e^{x} \cot \left(e^{x}\right) \csc \left(e^{x}\right) d x$$
Evaluate the integral.$$\int_{1}^{4} \frac{x^{2}+1}{\sqrt{x}} d x$$
Evaluate the integral.$$\int_{-2}^{0} x e^{-x^{2}} d x$$
You are given a pair of integrals. Evaluate the integral that can be worked using the techniques covered so far (the other cannot).$$\int \frac{5}{3+x^{2}} d x \quad \text { and } \quad \, \int \frac{5}{3+x^{3}} d x$$
You are given a pair of integrals. Evaluate the integral that can be worked using the techniques covered so far (the other cannot).$$\int \sin 2 x \, d x \quad \text { and } \quad \, \int \sin ^{2} x d x$$
You are given a pair of integrals. Evaluate the integral that can be worked using the techniques covered so far (the other cannot).$$\int \ln x \, d x \quad \text { and } \quad \, \int \frac{\ln x}{2 x} d x$$
You are given a pair of integrals. Evaluate the integral that can be worked using the techniques covered so far (the other cannot).$$\int \frac{x^{3}}{1+x^{8}} d x \quad \text { and } \quad \, \int \frac{x^{4}}{1+x^{8}} d x$$
You are given a pair of integrals. Evaluate the integral that can be worked using the techniques covered so far (the other cannot).$$\int e^{-x^{2}} d x \quad \text { and } \quad \int x e^{-x^{2}} d x$$
You are given a pair of integrals. Evaluate the integral that can be worked using the techniques covered so far (the other cannot).$$\int \sec x \, d x \quad \text { and } \, \int \sec ^{2} x d x$$
Find $\int_{0}^{2} f(x) d x,$ where $f(x)=\left\{\begin{array}{ll}x /\left(x^{2}+1\right) & \text { if } x \leq 1 \\ x^{2} /\left(x^{2}+1\right) & \text { if } x>1\end{array}\right.$
$$\text { Find } \int_{-2}^{2} f(x) d x, \text { where } f(x)=\left\{\begin{array}{ll}x e^{x^{2}} & \text { if } x<0 \\x^{2} e^{x^{3}} & \text { if } x \geq 0\end{array}\right.$$
Rework example 1.5 by rewriting the integral as$\int \frac{4 x+4}{2 x^{2}+4 x+10} d x-\int \frac{3}{2 x^{2}+4 x+10} d x$ and complet-ing the square in the second integral.