The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ e^(z + eᶻ) dz

Use any method to evaluate the integrals in Exercises 55–66.

∫ x / (x + √(x² + 2)) dx

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Use numerical integration to estimate the value of

arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).

For reference, arcsin(0.6) = 0.64350 to five decimal places.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ dx / (4 - x²)^(3/2) from 0 to 1

Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₁^∞ dx / [x√(x² − 1)]

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (x² / (x² + 1)) dx

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ sin(t / 3) sin(t / 6) dt

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ θ cos(πθ) dθ

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

∫ √x e√x dx

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.

85. Find the volume of the solid generated by revolving the region about the y-axis.

Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].

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 2 to ∞ of (dx / √(x² - 1))

Evaluate the integrals in Exercises 33–52.

∫ sec⁴(x) tan²(x) dx

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (8x² + 8x + 2) / (4x² + 1)² dx

Evaluate the integrals in Exercises 53–58.

∫ from -π/2 to π/2 of cos(x) cos(7x) dx

In Exercises 67–73, use integration by parts to establish the reduction formula.

∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx

Evaluate the integrals in Exercises 33–52.

∫ sec⁶(x) dx

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (dθ / cos θ - 1)

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ √(x - 2) / √(x - 1) dx

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ cos(θ / 2) cos(7θ) dθ

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)

Solve the initial value problems in Exercises 53–56 for y as a function of x.

(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ x² sin(x) dx

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

∫ 3 sec^4(3x) dx

Use any method to evaluate the integrals in Exercises 65–70.

∫ sin³(x) / cos⁴(x) dx

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

∫ (sin⁻¹ x)² / √(1 - x²) dx

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (s⁴ + 81) / (s(s² + 9)²) ds