CPM Homework Help : APCALC Problem 10-86

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10-86.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method.

$\int \operatorname { sin } ^ { 2 } ( x ) \operatorname { cos } ^ { 2 } ( x ) d x$
Step 1(a):
$\sin^2(x)\cos^2(x)=\frac{1}{4}\sin^2(2x)$
Step 2(a):
$=-\frac{1}{8}(2\sin^2(2x))$
Step 3(a):
$=-\frac{1}{8}(-1+(1-2\sin^2(2x)))$
Step 4(a):
$=-\frac{1}{8}(-1+\cos(4x))$
Rewrite the integrand using this final expression. You should now be able to complete the integration.

$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { m - 1 } } d m$
Partial Answer (b):
$=\left.\frac{\sqrt{m-1}}{1/2}\right|_1^3$

$\int \frac { x ^ { 3 } - 4 x ^ { 2 } + 7 } { x } d x$
Step 1(c):
$=\int\Big({x^2-4x+\frac{7}{x}}\Big)dx$

$\int \frac { \operatorname { cos } ( x ) } { 1 + \operatorname { sin } ^ { 2 } ( x ) } d x$
Hint (d):
Use substitution. Let $u = \sin(x)$ .
More Help (d):
Once the substitution is made, you should recognize the integrand as the derivative of an inverse trigonometric function.

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