7. Antiderivatives & Indefinite Integrals

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

∫ sin(2x) cos(3x) dx

Use the substitution u = tan x to evaluate the integral

∫ dx / (1 + sin² x).

9–61. Trigonometric integrals Evaluate the following integrals.

20. ∫ sin⁻³ᐟ²x cos³x dx

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ dt / (tan(t)√4 - sin^2(t))

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.

∫ csc θ dθ

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

∫ 3 sec^4(3x) dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ dx / (1 - sin² x)

9–61. Trigonometric integrals Evaluate the following integrals.

25. ∫ sin²x cos⁴x dx

9–61. Trigonometric integrals Evaluate the following integrals.

11. ∫ sin²(3x) dx

Evaluate the integrals in Exercises 33–52.

∫ 8 cot⁴(t) dt

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

68. ∫ sin(5x)sin(7x) dx

7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

8. ∫ sin 3x cos 2x dx

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

70. ∫ cos(x)cos(2x) dx

Evaluate ∫ sec θ dθ by:

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