Example 5: Trigonometric Substitution

The basic integrals:

(a)

(b)

(c) , x > 1

can be evaluated very easily by a trigonometric substitution,

(a)

Figure 7.6.6 

Let θ = arcsin x (Figure 7.6.6). Then

x = sin θ, dx = cos θ dθ, = cos θ.

(b)

Figure 7.6.7

Let θ = arctan x (Figure 7.6.7). Then

x = tan θ, dx = sec² θ dθ, = sec θ.

(c) , x > 1.

Figure 7.6.8

Let θ = arcsec x (Figure 7.6.8). Then

x = sec θ, dx = tan θ secθ dθ, = tan θ.

It is therefore more important to remember the method of trigonometric substitution than to remember the integration formulas (a), (b), (c).