### Home > APCALC > Chapter 9 > Lesson 9.4.3 > Problem9-133

9-133.

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. $\int \frac { 1 } { y ( 2 - y ) } d y$

Use partial fraction decomposition to rewrite the integrand.

$\frac{1}{y(2-y)}=\frac{a}{y}+\frac{b}{2-y}$

1. $\int \operatorname { sec } ( m ) \operatorname { tan } ( m ) \operatorname { ln } ( \operatorname { sec } ( m ) )$

Use substitution. Let $u = \sec(m)$.

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

Use partial fraction decomposition to rewrite the integrand.

$\frac{3}{x^2+4x+3}=\frac{a}{x+3}+\frac{b}{x+1}$

1. $\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } } d x$

This is an improper integral. Use a limit to properly evaluate this integral.