Math 152 – Spring 2016
Section 8.2
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Section 8.2 – Trigonometric Integrals
Example 1. Evaluate the following integral:
R
sin3 x dx
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How to integrate sinm x cosn x dx:
• If the power of cosine is odd (n = 2k + 1), then save one cosine factor and use
cos2 x = 1 − sin2 x to change the rest of the cosines to sines:
Z
Z
k
sinm x cos2k+1 x dx = sinm x cos2 x cos x dx
Z
k
= sinm x 1 − sin2 x cos x dx
Then do a subsitution with u = sin x and du = cos xdx
• If the power of sine is odd (m = 2k + 1), then save one sine factor and use
sin2 x = 1 − cos2 x to change the rest of the sines to cosines:
Z
Z
k
sin2k+1 x cosn x dx =
sin2 x cosn x sin x dx
Z
k
=
1 − cos2 x cosn x sin x dx
Then do a subsitution with u = cos x and du = − sin xdx
Example 2. Solve the following integrals.
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(a) 4 sin3 x cos8 x dx
Math 152 – Spring 2016
(b)
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cot5 x sin2 x dx
(c)
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1
x
Section 8.2
sin2 (ln x) cos3 (ln x) dx
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What if m and n are even in sinm x cosn x dx?
Then use the half-angle identities:
sin2 x =
1
(1 − cos 2x)
2
cos2 x =
1
(1 + cos 2x)
2
or the double-angle formula
sin x cos x =
Example 3. Solve the following integrals.
R π/2
(a) π/6 cos2 (2x) dx
1
sin 2x
2
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Math 152 – Spring 2016
(b)
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Section 8.2
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cos2 (3x) sin2 (3x) dx
(c) Find the volume of the region found by rotating the area bounded by y = cos x,
y = 0, x = 0, and x = π/2 about the line y = −1.
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How to integrate tanm x secn x dx:
• If the power of secant is even (n = 2k = 2(k − 1) + 2), save a factor of sec2 x and
use sec2 x = 1 + tan2 x to express the remaining factors in terms of tan x:
Z
Z
k−1
m
2k
tan x sec x dx = tanm x sec2 x
sec2 x dx
Z
k−1
= tanm x 1 + tan2 x
sec2 x dx
Then substitute u = tan x and du = sec2 x dx.
• If the power of tan x is odd (m = 2k + 1), then save a factor of sec x tan x and
use tan2 x = sec2 x − 1 to express the remaining factors in terms of sec x.
Z
Z
k
2k+1
n
tan
x sec x dx =
tan2 x secn−1 x sec x tan x dx
Z
k
=
sec2 x − 1 secn−1 x sec x tan x dx
Then substitute u = sec x and du = sec x tan x dx.
Math 152 – Spring 2016
Section 8.2
Example 4. Evaluate the following integrals.
R
(a) tan3 x sec5 x dx
x
2
x
2
(b)
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tan5
(c)
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8ex tan2 (ex ) dx
sec4
dx
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Math 152 – Spring 2016
(d)
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sec x dx
(e)
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sec3 x dx
Section 8.2
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How to integrate cotm x cscn x dx:
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Use the same rules as tanm x secn x dx replacing tan with cot and sec with csc.
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Example 5. Evaluate cot3 x csc6 x dx.
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