Integrals with Exponentials

Indefinite integrals of expressions containing exponentials.

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

\int x e^{a x} d x = \frac{1}{a} x e^{a x} + \frac{i π}{2 a ^{3/2}} erf (i a x), + C

where erf (x) = \frac{2}{π} \int_{0}^{x} e^{- t^{2}} d t

\int x e^{x} d x = (x - 1) e^{x} + C

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

\int x^{2} e^{x} d x = (x^{2} - 2 x + 2) e^{x} + C

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

\int x^{3} e^{x} d x = (x^{3} - 3 x^{2} + 6 x - 6) e^{x} + C

\int x^{n} e^{a x} d x = \frac{x ^{n} e ^{a x}}{a} - \frac{n}{a} \int x^{n - 1} e^{a x} d x

\int x^{n} e^{a x} d x = \frac{( - 1 ) ^{n}}{a ^{n + 1}} Γ [1 + n, - a x], + C

where Γ (a, x) = \int_{x}^{\infty} t^{a - 1} e^{- t} d t

\int e^{a x^{2}} d x = - \frac{i π}{2 a} erf (i x a) + C

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

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

\int x^{2} e^{- a x^{2}} d x = \frac{1}{4} \frac{π}{a ^{3}} erf (x a) - \frac{x}{2 a} e^{- a x^{2}} + C