Trigonometric Substitutions

Integrals containing one of the terms

, , or

can often be integrated by a trigonometric substitution. The idea is to take x, a, and the square root as the three sides of a right triangle and use one of its acute angles as a new variable θ. The three kinds of trigonometric substitutions are shown in Figure 7.6.1. These figures do not have to be memorized. Just remember that the sides must be labeled so that

(opposite)² + (adjacent)² = (hypotenuse)².

These substitutions frequently give an integral of powers of trigonometric functions discussed in the preceding section.

Figure 7.6.1

Example 1

Example 2

Example 3

Example 4

Example 5