### Home > APCALC > Chapter 9 > Lesson 9.1.3 > Problem9-42

9-42.

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. Homework Help ✎

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

$=\int_0^1\frac{1}{\sqrt{1-x^2}}dx+\int_0^1\frac{x}{\sqrt{1-x^2}}dx$

You should have the first integral committed to memory.
For the second integral, let $u = 1 - x^2$.

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

This is an improper integral because the graph has a vertical asymptote at $x = 0$. Rewrite this integral using limits and two separate integrals before evaluating.

1. $\int _ { 0 } ^ { 1 } \frac { 1 - x } { x } d x$

$=\int_0^1\Big(\frac{1}{x}-1\Big)dx$

1. $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 3 ] { x } } d x$

Rewrite the integrand using exponents. This is also an improper integral.