Problems

01

Distinct linear factors

Compute the integral:

$$\int \frac{1}{(x+2)(x-3)}dx$$

02

Long division first

Compute the integral:

$$\int \frac{2x^3+3x^2+7x+4}{x+1}dx$$

03

Repeated factor

Compute the integral:

$$\int \frac{1}{x{(x-1)}^2}dx$$

04

Partial fractions - irreducible quadratic

Compute the integral:

$$\int \frac{x^2}{x^2+9}dx$$

05

Partial fractions - long division

Compute the integral:

$$\int \frac{x^2-x+1}{x^2+x}dx$$

06

Partial fractions - big generic

Give the generic partial fraction decomposition (no need to solve for the constants):

$$\frac{x+2}{(x^2+2){(x-1)}^3(x^2-9)}=\frac{A}{?}+?$$

07

Partial fractions - linear and quadratic

Compute the integral:

$$\int \frac{5x^2-5x+14}{(x-2)(x^2+4)}dx$$

08

Partial fractions - repeated factor

Compute the integral:

$$\int \frac{1}{x{(x-1)}^3}dx$$

09

Partial fractions - rationalize first

For each of these integrals, make a $u$-substitution that changes the integrand into a rational function. Write the integral in terms of $u$ for your answer. You do not have to compute the $u$-integral.

(a) $\int \frac{1}{e^x-1}dx$

(b) $\int \frac{\sqrt{x}}{x-1}dx$