Integrals of Rational Functions

Indefinite integrals of rational expressions.

\int \frac{1}{( x + a ) ^{2}} d x = - \frac{1}{x + a} + C

\int (x + a)^{n} d x = \frac{( x + a ) ^{n + 1}}{n + 1} + C, n \neq = - 1

\int x (x + a)^{n} d x = \frac{( x + a ) ^{n + 1} (( n + 1 ) x - a )}{( n + 1 ) ( n + 2 )} + C

\int \frac{1}{1 + x ^{2}} d x = arctan x + C

\int \frac{1}{a ^{2} + x ^{2}} d x = \frac{1}{a} arctan \frac{x}{a} + C

\int \frac{x}{a ^{2} + x ^{2}} d x = \frac{1}{2} ln ∣ a^{2} + x^{2} ∣ + C

\int \frac{x ^{2}}{a ^{2} + x ^{2}} d x = x - a arctan \frac{x}{a} + C

\int \frac{x ^{3}}{a ^{2} + x ^{2}} d x = \frac{1}{2} x^{2} - \frac{1}{2} a^{2} ln ∣ a^{2} + x^{2} ∣ + C

\int \frac{1}{a x ^{2} + b x + c} d x = \frac{2}{4 a c - b ^{2}} arctan \frac{2 a x + b}{4 a c - b ^{2}} + C

\int \frac{1}{( x + a ) ( x + b )} d x = \frac{1}{b - a} ln \frac{a + x}{b + x}, a \neq = b + C

\int \frac{x}{( x + a ) ^{2}} d x = \frac{a}{a + x} + ln ∣ a + x ∣ + C

\int \frac{x}{a x ^{2} + b x + c} d x = \frac{1}{2 a} ln ∣ a x^{2} + b x + c ∣ - \frac{b}{a 4 a c - b ^{2}} arctan \frac{2 a x + b}{4 a c - b ^{2}} + C