Use any method to evaluate the integrals in Exercises 55–66.
∫ x / (x + √(x² + 2)) dx

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ √x e√x dx

Evaluate the integrals in Exercises 33–52.

∫ sec⁴(x) tan²(x) dx

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to 1 of ((e^(-√x)) / √x dx)

Equations (4) and (5) lead to different formulas for the integral of arctan x:

a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]

b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]

Can both integrations be correct? Explain.

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)

To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.

∫ log₂ x dx

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 8 cos^4(2πt) dt