Chapter 6

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{3 x^{2}-2 x+1}{2 x^{2}+3}\)

Find the integral. $$ \int \frac{1}{\sqrt{16-x^{2}}} d x $$

Use partial fractions to find the integral. $$ \int \frac{x}{16 x^{4}-1} d x $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int\left(x^{2}-1\right) e^{x} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty}(x-1) e^{-x} d x $$

Use Wallis's Formulas to evaluate the integral. $$ \int_{0}^{\pi / 2} \sin ^{7} x d x $$

Use integration tables to find the integral. $$ \int \frac{e^{x}}{\left(1-e^{2 x}\right)^{3 / 2}} d x $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{x+2}\)

Find the integral. $$ \int x \sqrt{16-4 x^{2}} d x $$

Use partial fractions to find the integral. $$ \int \frac{x^{2}-4 x+7}{x^{3}-x^{2}+x+3} d x $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int \frac{x}{\sqrt{2+3 x}} d x $$

Find the integral involving secant and tangent. $$ \int \sec ^{4} 5 x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} e^{-x} \cos x d x $$

Use integration tables to find the integral. $$ \int \frac{x}{\left(x^{2}-6 x+10\right)^{2}} d x $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{3}}{e^{x / 2}}\)

Find the integral. $$ \int \frac{1}{\sqrt{x^{2}-9}} d x $$

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{1} \frac{3}{2 x^{2}+5 x+2} d x $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x \cos x d x $$

Find the integral involving secant and tangent. $$ \int \sec ^{6} 3 x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} e^{-a x} \sin b x d x, \quad a>0 $$

Use integration tables to find the integral. $$ \int \frac{\cos x}{\sqrt{\sin ^{2} x+1}} d x $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{2}}{e^{x}}\)

Find the integral. $$ \int \frac{t}{\left(1-t^{2}\right)^{3 / 2}} d t $$

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{5} \frac{x-1}{x^{2}(x+1)} d x $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x \sin x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{4}^{\infty} \frac{1}{x(\ln x)^{3}} d x $$

Find the integral involving secant and tangent. $$ \int \sec ^{3} \pi x d x $$

Use integration tables to find the integral. $$ \int \frac{x^{3}}{\sqrt{4-x^{2}}} d x $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\)

Find the integral. $$ \int \frac{\sqrt{1-x^{2}}}{x^{4}} d x $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x^{3} \sin x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{\ln x}{x} d x $$

Find the integral involving secant and tangent. $$ \tan ^{2} x d x $$

Use integration tables to find the integral. $$ \int \tan ^{3} \theta d \theta $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{x^{2}}{\sqrt{x^{2}+1}}\)

Find the integral. $$ \int \frac{1}{x \sqrt{4 x^{2}+16}} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} \frac{2}{4+x^{2}} d x $$

Find the integral involving secant and tangent. $$ \int \tan ^{5} \frac{x}{4} d x $$

In Exercises 23-26, use integration tables to evaluate the integral. $$ \int_{0}^{1} x e^{x^{2}} d x $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\cos x}{x}\)

Find the integral. $$ \int \frac{-5 x}{\left(x^{2}+5\right)^{3 / 2}} d x $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int \arctan x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \frac{x^{3}}{\left(x^{2}+1\right)^{2}} d x $$

Find the integral involving secant and tangent. $$ \int \tan ^{3} \frac{\pi x}{2} \sec ^{2} \frac{\pi x}{2} d x $$

Use integration tables to evaluate the integral. $$ \int_{0}^{\pi} x \sin x d x $$

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\sin x}{x-\pi}\)

Find the integral. $$ \int(x+1) \sqrt{x^{2}+2 x+2} d x $$

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int e^{x} \cos 2 x d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \frac{1}{e^{x}+e^{-x}} d x $$

Find the integral involving secant and tangent. $$ \int \sec ^{2} x \tan x d x $$