### Home > CALC > Chapter 9 > Lesson 9.1.1 > Problem9-9

9-9.

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. $\int _ { - 1 } ^ { 1 } \frac { 3 y } { y ^ { 4 } + 1 } d y$

Graph the function. What do you notice?

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

Use partial fraction decomposition to rewrite the integrand as a sum of fractions.

$\frac{2}{y^2-1}=\frac{A}{y-1}+\frac{B}{y+1}$

1. $\int _ { 0 } ^ { \sqrt { 2 } / 2 } \frac { \operatorname { cos } ^ { - 1 } x } { \sqrt { 1 - x ^ { 2 } } } d x$

Use substitution.

Let $u =\operatorname{cos}^{–1}(x)$. Then $du =$ _____.

1. $\int \frac { d t } { \operatorname { cos } ^ { 2 } t - \operatorname { sin } ^ { 2 } t }$

Recall that $\operatorname{cos}^2(x)-\operatorname{sin}^2(x) =\operatorname{cos}(2x).$

$\int\frac{1}{\cos(2t)}dt=\int\sec(2t)dt$

$=\frac{1}{2}\int\sec(u)du\text{ }u=2x$

$=\frac{1}{2}\ln(\tan(u)+sec(u))$