### Home > APCALC > Chapter 11 > Lesson 11.3.1 > Problem11-82

11-82.

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. Homework Help ✎

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. $\int \sec^2(x) \ln(\tan(x))dx$

Let $u = \tan\left(x\right)$.

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. $\int 6x \tan(x^2)dx$

Use substitution twice.

Let $u = x^{2}$. Then $du = 2x$.

$= \int 3\tan(u)du$

Let $v = \cos\left(u\right)$. Then $dv = -\sin\left(u\right)du$.

$=-3\int\frac{1}{v}dv$