CPM Homework Help : APCALC Problem 12-21

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.

$\int \frac { 1 } { x \operatorname { ln } ( 3 x ) } d x$

$\int e ^ { x } \operatorname { sin } ( 2 x ) d x$
Step 1 (b):
Use integration by parts. Let $f = \sin(2x)$ and $dg = e^xdx$ .
Then $df = 2\cos(2x)dx$ and $g = e^{x}$ .
Step 2 (b):
$=e^x\sin(2x)-2\int e^x\cos(2x)dx$
Step 3 (b):
Use integration by parts on the integral in Step 2.
Let $f = \cos(2x)$ and $dg = e^{x}dx$ .
Then $df = -2\sin(2x)dx$ and $g = e^{x}$ .
Step 4 (b):
$\int e^x\sin(2x)=e^x\sin(2x)-2\Big(e^x\cos(2x)-(-2)\int e^x\sin(2x)dx\Big)$
Step 5 (b):
Use algebra to get the original integral expression by itself.

$\int \frac { d x } { x ^ { 2 } - x }$
Hint (c):
Use partial fraction decomposition.
$\frac{1}{x^2-x}=\frac{A}{x}+\frac{B}{x-1}$

$\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } d x$
Step 1 (d):
$\sqrt{4-x^2}=\sqrt{4(1-\frac{x^2}{4})}$
Step 2 (d):
$=2\sqrt{1-\Big(\frac{x}{2}\Big)^2}$
Step 3 (d):
Rewrite the denominator of the integrand using Steps 1 and 2.
You should recognize the integrand as a form of the derivative of a special function.