Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions

By ammar al-qadasi

Multiple Choice

$$\int\frac{dx}{x^2+2x-15}$$  

• $$\frac{1}{8}\ln\left|x-3\right|-\frac{1}{8}\ln\left|x+5\right|+c$$  

• $$\frac{1}{4}\ln\left|x-3\right|-\frac{1}{8}\ln\left|x+5\right|+c$$  

• $$\frac{1}{8}\ln\left|x+5\right|-\frac{1}{8}\ln\left|x-3\right|+c$$  

• $$\frac{1}{4}\ln\left|x+5\right|-\frac{1}{4}\ln\left|x-3\right|+c$$  

Multiple Choice

$$\int\frac{xdx}{\left(x-2\right)\left(x+1\right)^2}$$  

• $$\frac{2}{9}\ln\left|x-2\right|-\frac{2}{9}\ln\left|x+1\right|+\frac{1}{3\left(x+1\right)}+c$$  

• $$\frac{2}{9}\ln\left|x+1\right|-\frac{2}{9}\ln\left|x-2\right|-\frac{1}{3\left(x+1\right)}+c$$  

• $$\frac{2}{9}\ln\left|x-2\right|-\frac{2}{9}\ln\left|x+1\right|-\frac{1}{3\left(x+1\right)}+c$$  

• $$\frac{2}{9}\ln\left|x-2\right|+\frac{2}{9}\ln\left|x+1\right|+\frac{1}{3\left(x+1\right)}+c$$  

Multiple Choice

Expand this integral: $$\int_{ }^{ }\frac{3x+11}{x^2-x-6}dx$$

• $$3x+11=A\left(x-2\right)-B\left(x-3\right)$$

• $$3x+11\left(x^2-x-6\right)=A\left(x^2-x-6\right)+B\left(x^2-x-6\right)$$

• $$3x+11=A\left(x+2\right)+B\left(x-3\right)$$

• $$3x+11=A\left(x+2\right)-B\left(x-3\right)$$

Multiple Choice

Simplify this equation using long division: $$\frac{x^3+2x^2-3x+4}{x-7}$$

• $$x^2+9x+60$$

• $$x^2+9x+60-\frac{424}{x-7}$$

• $$x^2+9x+60+\frac{424}{x-7}$$

• $$x^2+9x+60+\frac{424}{x^2+9x+60}$$

Multiple Choice

Evaluate using partial fractions: $$\int_{ }^{ }\frac{x^2}{x^2-1}dx$$

• $$x+\frac{1}{2}\ln\left|x-1\right|-\frac{1}{2}\ln\left|x+1\right|+C$$

• $$x+\frac{1}{2}\ln\left|x-1\right|+\frac{1}{2}\ln\left|x+1\right|+C$$

• $$x+\ln\left|x-1\right|+C$$

• $$x^2+\frac{1}{2}\ln\left|x^2-1\right|-\frac{1}{2}\ln\left|x^2+1\right|+C$$