Chapter 7

Use substitution to evaluate the definite integrals. $$ \int_{1}^{9} \frac{1}{\sqrt{x}} e^{-\sqrt{x}} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{x^{2}+3} d x $$

Use substitution to evaluate the definite integrals. $$ \int_{0}^{2} x \sqrt{4-x^{2}} d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x}{x^{2}+3} d x $$

Use the fact that $$ \cot x=\frac{\cos x}{\sin x} $$ to evaluate $$ \int \cot x d x $$

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x+2}{x^{2}+2} d x $$

The integral $$ \int \ln x d x $$ can be evaluated in two ways. (a) Write \(\ln x=1 \cdot \ln x\) and use integration by parts to evaluate the integral. (b) Use the substitution \(u=\ln x\) and integration by parts to evaluate the integral.

Use an appropriate substitution followed by integration by parts to evaluate $$ \int x^{3} e^{-x^{2} / 2} d x $$

Use an appropriate substitution to evaluate $$ \int x(x-2)^{1 / 4} d x $$

Simplify the integrand and then use an appropriate substitution to evaluate $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{(\sin x-\cos x)^{2}} d x $$

Evaluate each definite integral. $$ \int_{1}^{4} e^{\sqrt{x}} d x $$

Evaluate each definite integral. $$ \int_{1}^{2} \ln \left(x^{2} e^{x}\right) d x $$

Evaluate each definite integral. $$ \int_{-1}^{0} \frac{2}{1+x^{2}} d x $$

Evaluate each definite integral. $$ \int_{1}^{2} x^{2} \ln x d x $$

Evaluate each definite integral. $$ \int_{0}^{\pi / 4} e^{x} \sin x d x $$

Evaluate each definite integral. $$ \int_{0}^{\pi / 6}\left(1+\tan ^{2} x\right) d x $$