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Problem 8.3.8
Evaluate the integrals in Exercises 1–22.
∫₀^π sin⁵(x/2) dx
Problem 8.3.66
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx
Problem 8.5.38
In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.
∫ 2y⁴ / (y³ - y² + y - 1) dy
Problem 8.3.46
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
Problem 8.5.70
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(t + 1) (dx/dt) = x² + 1 (for t > -1), x(0) = 0
Problem 8.3.68
Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx
Problem 8.5.4
Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)
Problem 8.1.10
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.
∫₁² (8 dx / (x² - 2x + 2))
Problem 8.3.26
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos²(θ)) dθ
Problem 8.3.38
Evaluate the integrals in Exercises 33–52.
∫ sec⁴(x) tan²(x) dx
Problem 8.5.52
Evaluate the integrals in Exercises 39–54.
∫ 1 / (cos θ + sin 2θ) dθ
Problem 8.5.68
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4
Problem 8.3.10
Evaluate the integrals in Exercises 1–22.
∫₀^(π/6) 3cos⁵(3x) dx
Problem 8.2.59
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 = ln(2) about the line x = ln(2).
Problem 8.1.40
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.
∫ (√x / (1 + x³)) dx
Hint: Let u = x^(3/2).
Problem 8.2.2
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ θ cos(πθ) dθ
Problem 8.1.42
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.
∫ ((2ˣ - 1) / 3ˣ) dx
Problem 8.5.64
Use any method to evaluate the integrals in Exercises 55–66.
∫ x / (x + √(x² + 2)) dx
Problem 8.3.18
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
Problem 8.4.42
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ dy / (y√(1 + (ln y)²)) from 1 to e
Problem 8.6.16
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ e^(-3t) sin(4t) dt
Problem 8.6.12
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(7 - x²))
Problem 8.6.40
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 √(2x - x^2) dx
Problem 8.1.24
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.
∫ (sec t + cot t)² dt
Problem 8.2.46
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ √x e√x dx
Problem 8.3.58
Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx
Problem 8.6.30
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ cos^(-1)(√x) / √x dx
Problem 8.8.68
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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))
Problem 8.2.30
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ z(ln z)² dz
Problem 8.5.62
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2 / (x(ln x - 2)³) dx