Home > CALC3RD > Chapter Ch9 > Lesson 9.1.1 > Problem9-9

9-9.

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

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

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

Recall that: $\cos^2(x) - \sin^2(x) = \cos(2x)$.

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

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

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