Difference between revisions of "009B Sample Final 3, Problem 5"

Revision as of 14:17, 28 February 2017

Find the following integrals.

(a) $\int x\cos(x)~dx$

(b) $\int \sin ^{3}(x)\cos ^{2}(x)~dx$

Foundations:
1. Integration by parts tells us that
$\int u~dv=uv-\int v~du.$
2. Since $\sin ^{2}x+\cos ^{2}x=1,$ we have
$\sin ^{2}x=1-\cos ^{2}x.$

Solution:

(a)

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

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 !Foundations: &nbsp;
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+|'''1.''' Integration by parts tells us that
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+|&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -12px">\int u~dv=uv-\int v~du.</math>
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+|'''2.''' Since &nbsp;<math style="vertical-align: -4px">\sin^2x+\cos^2x=1,</math>&nbsp; we have
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+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\sin^2x=1-\cos^2x.</math>
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