Chapter 14
Trigonometric Integrals, Trigonometric
Substitution, and Partial Fractions
T
his chapter covers trigonometric integrals, trigonometric substitutions, and partial
fractions — the remaining integration techniques you encounter in a second-semester
calculus course (in addition to u-substitution and integration by parts; see Chapter 13). In a
sense, these techniques are nothing fancy. For the trigonometric integrals, you typically use
a u-substitution followed by a trigonometric identity, possibly throwing in a bit of algebra to
solve the problem. For the trigonometric substitutions, you’re often integrating a function
involving a radical; by picking a clever substitution, you can often remove the radical and
make the problem into a trigonometric integral and proceed from there. Last, the partial
fractions technique simply decomposes a rational function into a bunch of simple fractions
that are easier to integrate.
With that said, many of these problems have many steps and require you to know identities,
polynomial long division, derivative formulas, and more. Many of these problems test your
algebra and trigonometry skills as much as your calculus skills.
The Problems You’ll Work On
This chapter finishes off the integration techniques that you see in a calculus class:
✓ Solving definite and indefinite integrals involving powers of trigonometric functions
✓ Solving definite and indefinite integrals using trigonometric substitutions
✓ Solving definite and indefinite integrals using partial fraction decompositions
Trigonometric Integrals, Trigonometric
Substitution, and Partial Fractions
T
his chapter covers trigonometric integrals, trigonometric substitutions, and partial
fractions — the remaining integration techniques you encounter in a second-semester
calculus course (in addition to u-substitution and integration by parts; see Chapter 13). In a
sense, these techniques are nothing fancy. For the trigonometric integrals, you typically use
a u-substitution followed by a trigonometric identity, possibly throwing in a bit of algebra to
solve the problem. For the trigonometric substitutions, you’re often integrating a function
involving a radical; by picking a clever substitution, you can often remove the radical and
make the problem into a trigonometric integral and proceed from there. Last, the partial
fractions technique simply decomposes a rational function into a bunch of simple fractions
that are easier to integrate.
With that said, many of these problems have many steps and require you to know identities,
polynomial long division, derivative formulas, and more. Many of these problems test your
algebra and trigonometry skills as much as your calculus skills.
The Problems You’ll Work On
This chapter finishes off the integration techniques that you see in a calculus class:
✓ Solving definite and indefinite integrals involving powers of trigonometric functions
✓ Solving definite and indefinite integrals using trigonometric substitutions
✓ Solving definite and indefinite integrals using partial fraction decompositions