Chapter 8

Evaluate. $$\int_{0}^{4} \frac{x^{3 / 2}}{x+1} d x$$

Calculate. $$\int_{0}^{1 / 4} \arcsin 2 x d x$$

Calculate. $$\int_{1}^{e} \frac{\ln x^{3}}{x} d x$$

Estimate the theoretical error if the trapezoidal rule with \(n=30\) is used to approximate $$\int_{2}^{7} \frac{x^{2}}{x^{2}+1} d x$$

Evaluate. $$\int_{1}^{3} \frac{1}{x^{3}+x} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \cot ^{5} 2 x d x$$.

Calculate. $$\int \frac{x+2}{\sqrt{x^{2}+4 x+13}} d x$$.

Evaluate. $$\int_{0}^{8} \frac{1}{1+\sqrt[3]{x}} d x$$

Calculate. $$\int_{0}^{\pi / 4} \frac{\arctan x}{1+x^{2}} d x$$

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

Evaluate. $$\int_{1}^{3} \frac{x^{2}-4 x+3}{x^{3}+2 x^{2}+x} d x$$

Calculate. $$\int \frac{x}{\left(x^{2}+2 x+5\right)^{2}} d x$$.

Evaluate. $$\int_{0}^{\pi / 2} \frac{\sin 2 x}{2+\cos x} d x$$

Calculate. $$\int_{0}^{1} x \arctan x^{2} d x$$

Calculate. $$\int \frac{\arcsin x}{\sqrt{1-x^{2}}} d x$$

Evaluate. $$\int_{0}^{2} \frac{x^{3}}{\left(x^{2}+2\right)^{2}} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int_{r}^{1 / 2} \cos \pi x \cos \frac{1}{2} \pi x d x$$.

Calculate. $$\int \frac{x}{\sqrt{x^{2}-2 x+3}} d x$$.

Evaluate. $$\int_{0}^{\pi / 2} \frac{1}{1+\sin x} d x$$

Calculate. $$\int e^{x} \cosh \left(2-e^{x}\right) d x$$

Calculate. $$\int \frac{\cos \theta}{\sin ^{2} \theta-2 \sin \theta-8} d \theta \cdot$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int_{0}^{z / 4} \cos 4 x \sin 2 x d x$$.

Use integration by parts to derive the formula. $$\int \operatorname{arcsec} x d x=x \operatorname{arcsec} x-\ln |x+\sqrt{x^{2}-1}|+C.$$

Evaluate. $$\int_{0}^{\pi / 3} \frac{1}{\sin x-\cos x-1} d x$$

Calculate. $$\int x^{2} \cos 112 x d x$$

Calculate. $$\int \frac{1}{x \ln x} d x$$

Calculate. $$\int \frac{e^{t}}{e^{2 t}+5 e^{t}+6} d t$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int(\sin 3 x-\sin x)^{2} d x$$.

$$\text { Calculate } \int \frac{1}{x} \sqrt{a^{2}-x^{2}} d x.$$ (a) by setting \(u=\sqrt{a^{2}-x^{2}}\) (b) by a trigonometric substitution. (c) Then reconcile the results.

Evaluate. $$\int_{0}^{1} \frac{\sqrt{x}}{1+\sqrt{x}} d x$$

Calculate. $$\int_{-1}^{1} x \sinh 2 x^{2} d x$$

Calculate. $$\int_{-1}^{1} \frac{x^{2}}{x^{2}+1} d x$$

Calculate. $$\int \frac{1}{\left.t(\ln t)^{2}-4\right)} d t$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \tan ^{4} x \sec ^{4} x d x$$.

Use the method of this section to show that $$ \int \sec x d x=\int \frac{1}{\cos x} d x=\ln \left|\frac{1+\tan \frac{1}{2} x}{1-\tan \frac{1}{2} x}\right|+C $$

Calculate. $$\int_{0}^{\pi / 4} \frac{1+\sin x}{\cos ^{2} x} d x$$

Calculate. $$\int \frac{1}{x} \arcsin (\ln x) d x$$

Calculate. $$\int \frac{\sec ^{2} \theta}{\tan ^{3} \theta-\tan ^{2} \theta} d \theta$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int \cot ^{4} x \csc ^{6} x d x$$.

Use \((8.4 .3)\) to calculate the integral.. $$\int \frac{1}{\left(x^{2}+1\right)^{2}} d x.$$

(a) Another way to calculate \(\int \sec x d x\) is to write $$ \int \sec x d x=\int \frac{\cos x}{\cos ^{2} x} d x=\int \frac{\cos x}{1-\sin ^{2} x} d x $$ Use the method of this section to show that $$ \int \sec x d x=\ln \sqrt{\frac{1+\sin x}{1-\sin x}}+C $$ (b) Show that the result in part (a) is equivalent to the familiar formula $$\int \sec x d x \quad \ln |\operatorname{scc} x| \tan x |+C$$

Calculate. $$\int \cos (\ln x) d x$$

Calculate. $$\int_{0}^{1 / 2} \frac{1+x}{\sqrt{1-x^{2}}} d x$$

Derive the formula.$$\int \frac{u}{a+b u} d u=\frac{1}{b^{2}}(a+b u-a \ln |a+b u|)+C$$.

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sin \frac{1}{2} x \cos 2 x d x$$.

Use \((8.4 .3)\) to calculate the integral.. $$\int \frac{1}{\left(x^{2}+1\right)^{3}} d x.$$

Calculate. $$\int \sin (\ln x) d x$$

Calculate using our table of integrals. $$\int \sqrt{x^{2}-4} d x$$

Derive the formula.$$\int \frac{d u}{u(a+b u)}=\frac{1}{a} \ln \left|\frac{u}{a+b u}\right|+C$$.

Calculate. (If you run out of ideas, use the examples as models.) $$\int_{0}^{2 \pi} \sin ^{2} a x d x, a \neq 0$$.