  ### Home > CALC > Chapter 7 > Lesson 7.4.4 > Problem7-207

7-207.

No calculator! 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. $\int \operatorname { cos } ^ { - 1 } x d x$

$f=\text{cos}^{-1}x, \ dg=dx, \ df=-\frac{1}{\sqrt{1-x^{2}}}dx, \ g=x$

$\int \text{cos}^{-1}xdx=x\text{cos}^{-1}x+\int \frac{xdx}{\sqrt{1-x^{2}}}$

Use integration by parts and substitution.

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

1. $\int _ { - 1 } ^ { 3 } \frac { d x } { x ^ { 2 / 3 } }$

There is a vertical asymptote at $x = 0$, so this improper integral needs to be rewritten in proper form (as a limit).

In fact, write a sum of two proper integrals.

1. $\int \operatorname { sec } ^ { 2 } x \operatorname { tan } x d x$

Use u-substitution. $\operatorname{sec}^2x\operatorname{tan}x = (\operatorname{sec}x)(\operatorname{sec}x)(\operatorname{tan}x)$ 