Download Section 7.2

Trigonometric Integrals - Section 7.2
Integrals Involving Powers of Sine and Cosine: ∫ sin m x cos n x dx
Useful trigonometric identities:
sin 2 x + cos 2 x = 1
tan 2 x + 1 = sec 2 x
sin 2x = 2 sin x cos x
sin 2 x = 1 1 − cos 2x
2
1 + cot 2 x = csc 2 x
cos 2x = cos 2 x − sin 2 x
cos 2 x = 1 1 + cos 2x
2
(Pythagorean Identities)
(Double Angle Identities)
(Power Reducing Formulas)
Use a trigonometric identity to evaluate the following integral:
∫ sin 2 x dx
Now evaluate the integral ∫ sin 3 x dx
1
Reduction Formulas for Integrals Involving Powers of Sine and Cosine
We can prove these using integration by parts
∫ sin n x dx = − 1n sin n−1 x cos x +
∫ cos n x dx =
n−1
n
1 cos n−1 x sin x + n − 1
n
n
∫ sin n−2 x dx
∫ cos n−2 x dx
General Guidelines for evaluating integrals of the form ∫ sin m x cos n x dx
1. If the power of sine is odd and positive, save one sine factor and convert the remaining factors to cosine by the
Pythagorean Identity. Then, expand and integrate by substitution choosing u = cos x.
∫ sin
Odd

2k+1
Convert to cosine
∫
x cos n x dx =
sin 2 x
k
Save for du
cos n x sin x dx = ∫1 − cos 2 x k cos n x sin x dx
2. If the power of cosine is odd and positive, save one cosine factor and convert the remaining factors to sine by
the Pythagorean Identity. Then, expand and integrate by substitution choosing u = sin x.
∫ sin m x cos
Odd

2k+1
Convert to sine
x dx =
∫ sin m x
cos 2 x
k
Save for du
cos x dx =
∫ sin m x1 − sin 2 x k cos x dx
3. If the powers of both sine and cosine are even and nonnegative, make repeated use of the identities
sin 2 x = 1 − cos 2x and cos 2 x = 1 + cos 2x
2
2
to convert the integrand to odd powers of the cosine. Then proceed as in guideline #2.
Alternatively, if m ≤ n, use the identity sin 2 x = 1 − cos 2 x to replace sin m x by 1 − cos 2 x m/2 to obtain
∫ sin m x cos n x dx = ∫1 − cos 2 x m/2 cos n x dx
and expand the integral on the right to obtain a sum of integrals of powers of cos x and use the reduction
formula
∫ cos n x dx =
If m > n, replace cos n x by 1 − sin 2 x
n/2
1 cos n−1 x sin x + n − 1
n
n
∫ cos n−2 x dx
to obtain
∫ sin m x cos n x dx = ∫ sin m x1 − sin 2 x n/2 dx
and expand the integral on the right to obtain a sum of integrals of powers sin x and evaluate using the reduction
formula
∫ sin n x dx = − 1n sin n−1 x cos x +
n−1
n
∫ sin n−2 x dx
2
Evaluate the following Integrals:
1.
∫ sin 2 x cos 3 x dx
2.
∫ sin 5 x cos 4 x dx
3
3.
∫ sin 6 x dx
4.
∫ sec x dx
4
Integrals Involving Powers of Tangent and Secant
Recall these integral forms:
∫ tan x dx = − ln|cos x| + C = ln|sec x| + C
∫ sec x dx = ln|sec x + tan x| + C
Use the following guidelines for integrals of the form ∫ sec m x tan n x dx (taken from Calculus by Larson, Hostetler,
& Edwards, 6th edition):
1. If the power of the secant is even and positive, save a secant-squared factor (for the du) and convert the
remaining factors to tangents. Then expand and integrate by substitution with u = tan x.
Convert to tangents
Even
∫ sec

2k
x tan n x dx =
∫
sec 2 x
Save for du
tan n x sec 2 x dx= ∫1 + tan 2 x k−1 tan n x sec 2 x dx
k−1
2. If the power of the tangent is odd and positive, save a secant-tangent factor (for the du) and convert the
remaining factors to secants. Then expand and integrate by substitution with u = sec x.
Convert to secants
Odd
∫ sec m x tan

2k+1
x dx =
∫ sec m−1 x
tan 2 x
k
Save for du
sec x tan x dx=
∫ sec m−1 xsec 2 x − 1 k sec x tan x dx
Evaluate the following:
For #’s 1 & 2 use the substitution u = tan x and reserve a sec 2 x factor for the "du"
1.
∫ tan 2 x sec 2 x dx
5
2.
∫ tan 2 x sec 4 x dx
Let u = sec x and reserve a sec x tan x factor for the "du"
1.
∫ tan 3 x sec 3 x dx
6