4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.

81. Find the values of p for which each integral converges.

a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ sinx·cos²x dx

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 2 of (t³ + t) dt

135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:

(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (z + 1) / [z²(z² + 4)] dz

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from -1 to 1 of (x² + 1) dx