### Home > APCALC > Chapter 9 > Lesson 9.2.1 > Problem9-55

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. Homework Help ✎

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

Use integration by parts.

Let $f = \ln(x)$ and $dg = x^3dx$.

1. $\int ( 2 x - 1 ) e ^ { x } d x$

Let $f = 2x − 1$ and $dg = e^xdx$.

1. $\int \frac { x ^ { 2 } - 4 x + 4 } { x + 2 } d x$

Use long division to rewrite the integrand.

1. $\int \frac { x + 1 } { x ( x ^ { 2 } + 1 ) } d x$

Use partial fraction decomposition to rewrite the integrand in the form:

$\frac{a}{x}+\frac{bx+c}{x^2+1}$

$=\frac{1}{x}+\frac{-x+1}{x^2+1}=\frac{1}{x}-\frac{x}{x^2+1}+\frac{1}{x^2+1}$