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

∫₀^(π/3) tan³x·sec²x dx

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.

∫ [y / √(16 − y²)] dy

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ

Evaluate the integrals in Exercises 37–44.

∫ cos⁵(x) sin⁵(x) dx

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

∫ (x + 1) / (x⁴ − x³) dx

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [(4x) / (x³ + 4x)] dx

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.

∫ x·sec²x dx

Evaluate the integrals in Exercises 33–36.

∫ [1 / √(9 - x²)] dx

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ x² ln(x) dx

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

123. ∫ √x * √(1 + √x) dx

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

129. ∫ (x^(ln x) * ln x) / x dx

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.

Consider the region bounded by the graphs of

y = ln(x), y = 0, and x = e.

a. Find the area of the region.

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 -2 to 0 of (x² - 1) dx

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.

a. About the y-axis.

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

a. u = 1/(x + 1)

What is the value of the integral?

Centroid:

Find the centroid of the region cut from the first quadrant by the curve

y = 1/√(x + 1) and the line x = 3.

Evaluate ∫ sec θ dθ by:

a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.

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

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

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 sin(x + 1) 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 1 to 2 of 1/s² ds

Finding area

Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:

a. π/2 ≤ x ≤ 3π/2.

Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

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 2 to 4 of 1/(s - 1)² ds

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 (t³ + 1) dt

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 3 of (2x - 1) dx

Evaluate ∫ x³ √(1 - x²) dx using:

a. Integration by parts.