91Ó°ÊÓ

Prove each statement in Exercises 74–77, using limits of definite integrals for general values of p.

If p > 1, then${∫}_1^{∞}\frac{1}{x^p}dx$converges to$\frac{1}{p-1}.$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

${∫}_2^3\frac{1}{x\sqrt{\ln x}}dx$

Solve the definite integral.

${∫}_{-1}^1{2}^x\cos xdx$

Solve each of the definite integrals in Exercises 67–76.

${∫}_0^{\mathrm{π}/4}\frac{\sin x}{{\cos }^{2}x}dx$.

Prove each statement in Exercises 74–77, using limits of definite integrals for general values of p.

If 0 < p < 1, then${∫}_0^1\frac{1}{x^p}dx$converges to$\frac{1}{1-p}.$

Solve each of the integrals in Exercises $75–78$by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.
$∫\frac{x^2-1}{x^2+1}dx$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

${∫}_2^4\frac{e^{2x}}{1+e^{2x}}dx$

Solve the definite integral.

${∫}_{\frac{\mathrm{π}}{4}}^{\frac{\mathrm{π}}{2}}xcs{c}^{2}xdx$

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

$∫\frac{x^4-3}{2+3x^2}dx$

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

${∫}_0^1\frac{x}{1+x^4}dx$