Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
c. u = arctan x
What is the value of the integral?

Evaluate the integrals in Exercises 91–102.

102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) dx

Verify the integration formulas in Exercises 37–40.

37. a. ∫sech(x)dx = tan⁻¹(sinh x) + C

Verify the integration formulas in Exercises 37–40.

39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C

Find the limits in Exercises 1–6.

1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)

Evaluate the integral.

${\int }_{}^{}(\frac{2}{\sqrt{1-x^2}}-\frac{1}{3\sqrt{x}})dx$

Evaluate the integral.

${\int }_{\sqrt{2}}^2\frac{1}{\mid x\mid \sqrt{x^2-1}}dx$

Find the indefinite integral.

${\int }_{}^{}\frac{1}{x\sqrt{4x^2-16}}dx$

Evaluate the definite integral in terms of an inverse trig function.

$\int_0^{\frac{\sqrt3}{3}}\frac{dx}{\sqrt{4-9x^2}}$