  ### Home > APCALC > Chapter 10 > Lesson 10.1.7 > Problem10-86

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.

1. $\int \operatorname { sin } ^ { 2 } ( x ) \operatorname { cos } ^ { 2 } ( x ) d x$

$\sin^2(x)\cos^2(x)=\frac{1}{4}\sin^2(2x)$

$=-\frac{1}{8}(2\sin^2(2x))$

$=-\frac{1}{8}(-1+(1-2\sin^2(2x)))$

$=-\frac{1}{8}(-1+\cos(4x))$

Rewrite the integrand using this final expression. You should now be able to complete the integration.

1. $\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { m - 1 } } d m$

$=\left.\frac{\sqrt{m-1}}{1/2}\right|_1^3$

1. $\int \frac { x ^ { 3 } - 4 x ^ { 2 } + 7 } { x } d x$

$=\int\Big({x^2-4x+\frac{7}{x}}\Big)dx$

1. $\int \frac { \operatorname { cos } ( x ) } { 1 + \operatorname { sin } ^ { 2 } ( x ) } d x$

Use substitution. Let $u = \sin(x)$.

Once the substitution is made, you should recognize the integrand as the derivative of an inverse trigonometric function.