CPM Homework Help : CALC Problem 10-108

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.

$\int \frac { 1 } { x ^ { 2 } + 4 } d x$
Hint (a):
You should recognize the integrand as a form of the derivative of $\operatorname{tan}^{–1}(x)$ .
The denominator needs to be of the form $a^2 + 1$ .
Step (a):
$\frac{1}{x^2+4}=\frac{1}{4(x^2/4+1)}=\frac{1}{4((x/2)^2+1)}$

$\int \frac { x } { x ^ { 2 } + 4 } d x$
Hint (b):
Use substitution. Let $u = x^2 + 4$ .

$\int \frac { 1 } { x ^ { 2 } + 4 x } d x$
Hint (c):
Factor the denominator then use partial fraction decomposition.

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