### Home > CALC > Chapter 11 > Lesson 11.3.1 > Problem11-97

11-97.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy. After evaluating each integral, write a short description of your method.

1. $\int _ { 0 } ^ { 2 } \frac { 1 } { ( x - 2 ) ^ { 2 } } d x$

The Reverse Chain Rule can be used here, but this is an improper integral, so be sure to use a limit.

1. $\int \operatorname { sec } ^ { 2 } x \operatorname { ln } ( \operatorname { tan } x ) d x$

Let $u =\operatorname{tan}(x)$.

1. $\int \frac { 3 } { ( x - 1 ) ( x + 2 ) } d x$

Use partial fraction decomposition.
$\frac{A}{x-1}+\frac{B}{x+2}=\frac{3}{(x-1)(x+2)}$

1. $\int 6 x \operatorname { tan } x ^ { 2 } d x$

Use substitution twice.

Let $u = x^2$. Then $du = 2x$.
$= \int 3\tan(u)du$

Let $v =\operatorname{cos}(u)$. Then $dv = –\operatorname{sin}(u)du$.
$=-3\int\frac{1}{v}dv$