CPM Homework Help : APCALC Problem 10-156

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10-156.

Examine the integrals below. Consider the multiple tools you have available for integrating and use the best one for each part. Evaluate each integral and briefly describe your method.

$\int _ { - \infty } ^ { \infty } \frac { 1 } { x ^ { 2 } + 25 } d x$
Step 1 (a):
$=\frac{1}{25}\int_{-\infty}^{\infty}\frac{1}{(x/5)^2+1}dx$
Step 2 (a):
$=\lim\limits_{a\to-\infty}\frac{1}{25}\int_{a}^{0}\frac{1}{(x/5)^2+1}dx+\lim\limits_{b\to\infty}\frac{1}{25}\int_{0}^{b}\frac{1}{(x/5)^2+1}dx$

$\int _ { 0 } ^ { 4 } \frac { x } { \sqrt { 16 - x ^ { 2 } } } d x$
Hint (b):
Use substitution. Let $u = 16 - x^2$ .
More Help (b):
A limit is needed to evaluate this integral.

$\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 3 } } d x$
Step (c):
$=\lim\limits_{c\to-\infty}\int_{c}^{-1}x^{-3}dx$

$\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x - 3 } d x$
Step (d):
$\frac{1}{x^2-2x-3}=\frac{-1/4}{x+1}+\frac{1/4}{x-3}$

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