CPM Homework Help : CALC Problem 9-43

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9-43.

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.

$\int _ { 0 } ^ { 1 } \frac { 1 + x } { \sqrt { 1 - x ^ { 2 } } } d x$
Step 1(a):
$=\int_0^1\frac{1}{\sqrt{1-x^2}}dx+\int_0^1\frac{x}{\sqrt{1-x^2}}dx$
Step 2(a):
You should have the first integral committed to memory.
For the second integral, let $u = 1 − x^2$ .

$\int _ { - \infty } ^ { \infty } \frac { 1 } { x } d x$
Hint (b):
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.

$\int _ { 0 } ^ { 1 } \frac { 1 - x } { x } d x$
Step (c):
$=\int_0^1\Big(\frac{1}{x}-1\Big)dx$

$\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 3 ] { x } } d x$
Hint (d):
Rewrite the integrand using exponents. This is also an improper integral.

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