CPM Homework Help : APCALC Problem 9-55

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.

$\int x ^ { 3 } \operatorname { ln } ( x ) d x$
Hint (a):
Use integration by parts.
Step (a):
Let $f = \ln(x)$ and $dg = x^3dx$ .

$\int ( 2 x - 1 ) e ^ { x } d x$
Step (b):
Let $f = 2x − 1$ and $dg = e^xdx$ .

$\int \frac { x ^ { 2 } - 4 x + 4 } { x + 2 } d x$
Hint (c):
Use long division to rewrite the integrand.

$\int \frac { x + 1 } { x ( x ^ { 2 } + 1 ) } d x$
Hint (d):
Use partial fraction decomposition to rewrite the integrand in the form:
$\frac{a}{x}+\frac{bx+c}{x^2+1}$
Step:
$=\frac{1}{x}+\frac{-x+1}{x^2+1}=\frac{1}{x}-\frac{x}{x^2+1}+\frac{1}{x^2+1}$