### Home > APCALC > Chapter 7 > Lesson 7.4.3 > Problem7-206

7-206.

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. $\int _ { - 1 } ^ { 3 } \frac { d x } { x }$

This is an improper integral because $y=\frac{1}{x}$has a vertical asymptote at $x=0$.

Rewrite properly (with limits) before evaluating.

1. $\int \operatorname { tan } ^ { - 1 } ( x ) d x$

Use integration by parts after you rewrite the integrand as $\arctan(x)$.
Let $f(x) = \arctan(x)$and $dg = dx$.
Then $df = \text{_______}$ and $g(x) = \text{_______}$.
Warning: You will have to use $U$-substitution to evaluate the integral.

1. $\quad \int \frac { \operatorname { sin } \sqrt { x } d x } { \sqrt { x } }$

$\text{Use }U\text{-Substitution}. \text{ Let }U=\sqrt{x}.$

1. $\int _ { \pi / 3 } ^ { \pi / 2 } \operatorname { csc } ^ { 2 } ( x ) d x$

$\text{Recall that }\frac{d}{dx}\cot(x)=-\csc^2(x).$