August 13, 2026 · Reference

Integrals with Radicals

Indefinite integrals of expressions containing radicals.

\int x - a d x = \frac{2}{3} (x - a)^{3/2} + C

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

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

\int x x - a d x = ⎩ ⎨ ⎧ \frac{2 a}{3} (x - a)^{3/2} + \frac{2}{5} (x - a)^{5/2}, or \frac{2}{3} x (x - a)^{3/2} - \frac{4}{15} (x - a)^{5/2}, or \frac{2}{15} (2 a + 3 x) (x - a)^{3/2} + C

\int a x + b d x = (\frac{2 b}{3 a} + \frac{2 x}{3}) a x + b + C

\int (a x + b)^{3/2} d x = \frac{2}{5 a} (a x + b)^{5/2} + C

\int \frac{x}{x \pm a} d x = \frac{2}{3} (x \mp 2 a) x \pm a + C

\int \frac{x}{a - x} d x = - x (a - x) - a arctan \frac{x ( a - x )}{x - a} + C

\int \frac{x}{a + x} d x = x (a + x) - a ln [x + x + a] + C

\int x a x + b d x = \frac{2}{15 a ^{2}} (- 2 b^{2} + ab x + 3 a^{2} x^{2}) a x + b + C

\int x (a x + b) d x = \frac{1}{4 a ^{3/2}} [(2 a x + b) a x (a x + b) - b^{2} ln a x + a (a x + b)] + C

\int x^{3} (a x + b) d x = [\frac{b}{12 a} - \frac{b ^{2}}{8 a ^{2} x} + \frac{x}{3}] x^{3} (a x + b) + \frac{b ^{3}}{8 a ^{5/2}} ln a x + a (a x + b) + C

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

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

\int x x^{2} \pm a^{2} d x = \frac{1}{3} (x^{2} \pm a^{2})^{3/2} + C

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

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

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

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

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

\int a x^{2} + b x + c d x = \frac{b + 2 a x}{4 a} a x^{2} + b x + c + \frac{4 a c - b ^{2}}{8 a ^{3/2}} ln 2 a x + b + 2 a (a x^{2} + b x + c) + C

\int x a x^{2} + b x + c d x = \frac{1}{48 a ^{5/2}} (2 a a x^{2} + b x + c (- 3 b^{2} + 2 ab x + 8 a (c + a x^{2})) + 3 (b^{3} - 4 ab c) ln b + 2 a x + 2 a a x^{2} + b x + c) + C

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

\int \frac{x}{a x ^{2} + b x + c} d x = \frac{1}{a} a x^{2} + b x + c - \frac{b}{2 a ^{3/2}} ln 2 a x + b + 2 a (a x^{2} + b x + c) + C

\int \frac{d x}{( a ^{2} + x ^{2} ) ^{3/2}} = \frac{x}{a ^{2} a ^{2} + x ^{2}} + C