Download Section 7.3

Trigonometric Substitution - Section 7.3
We now look at a technique for evaluating integrals involving square root expressions of the form:
a2 − x2
x2 + a2
x2 − a2
where a > 0 is a constant.
Useful trigonometric identities:
sin 2 x + cos 2 x = 1
tan 2 x + 1 = sec 2 x
1 + cot 2 x = csc 2 x
(Pythagorean Identities)
Evaluate the following integral:
∫
1 − x 2 dx
1
Integrals involving a 2 − x 2
If a 2 − x 2 occurs in the integral where a > 0, try the substitution
x = a sin θ with − π ≤ θ ≤ π
2
2
so that
dx = a cos θ dθ
and
a2 − x2 =
a 2 − a 2 sin 2 θ = a 1 − sin 2 θ = a cos 2 θ = a cos θ
Example: Evaluate the following integral
∫
x2
dx
3/2
4 − x 2 
2
Integrals involving x 2 + a 2
If x 2 + a 2 occurs in the integral where a > 0, try the substitution
x = a tan θ with − π < θ < π
2
2
so that
dx = a sec 2 θ dθ
and
x2 + a2 =
a 2 tan 2 θ + a 2 = a tan 2 θ + 1 = a sec 2 θ = a sec θ
Example: Evaluate the following integral
∫
4t 2 + 12 dt
3
Integrals involving x 2 − a 2
If x 2 − a 2 occurs in the integral where a > 0, try the substitution
x = a sec θ with 0 ≤ θ < π if x ≥ a or π ≤ θ < 3π if x ≤ −a
2
2
so that
dx = a sec θ tan θ dθ
and
x2 − a2 =
a 2 sec 2 θ − a 2 = a sec 2 θ − 1 = a tan 2 θ = a tan θ
Example: Evaluate the following integral
∫
x
2
dx
x2 − 9
4
Completing the Square with Trigonometric Substitution
Evaluate the following integral using trigonometric substitution
∫
dx
2
x 2 − 6x + 13
5
Use trigonometric substitution to evaluate the integral needed to find the area enclosed by a circle with radius r
6