List of integrals of exponential functions

Indefinite integral

Indefinite integrals are antiderivative functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.

Integrals involving polynomials

∫ x e c x d x = e c x ( c x − 1 c 2 ) ∫ x 2 e c x d x = e c x ( x 2 c − 2 x c 2 + 2 c 3 ) ∫ x n e c x d x = 1 c x n e c x − n c ∫ x n − 1 e c x d x = ( ∂ ∂ c ) n e c x c = e c x ∑ i = 0 n ( − 1 ) i n ! ( n − i ) ! c i + 1 x n − i = e c x ∑ i = 0 n ( − 1 ) n − i n ! i ! c n − i + 1 x i ∫ e c x x d x = ln ⁡ | x | + ∑ n = 1 ∞ ( c x ) n n ⋅ n ! ∫ e c x x n d x = 1 n − 1 ( − e c x x n − 1 + c ∫ e c x x n − 1 d x ) (for  n ≠ 1 )

Integrals involving only exponential functions

∫ f ′ ( x ) e f ( x ) d x = e f ( x ) ∫ e c x d x = 1 c e c x ∫ a c x d x = 1 c ⋅ ln ⁡ a a c x f o r a > 0 ,   a ≠ 1

Integrals involving exponential and trigonometric functions

∫ e c x sin ⁡ b x d x = e c x c 2 + b 2 ( c sin ⁡ b x − b cos ⁡ b x ) = e c x c 2 + b 2 sin ⁡ ( b x − ϕ ) cos ⁡ ( ϕ ) = c c 2 + b 2 ∫ e c x cos ⁡ b x d x = e c x c 2 + b 2 ( c cos ⁡ b x + b sin ⁡ b x ) = e c x c 2 + b 2 cos ⁡ ( b x − ϕ ) cos ⁡ ( ϕ ) = c c 2 + b 2 ∫ e c x sin n ⁡ x d x = e c x sin n − 1 ⁡ x c 2 + n 2 ( c sin ⁡ x − n cos ⁡ x ) + n ( n − 1 ) c 2 + n 2 ∫ e c x sin n − 2 ⁡ x d x ∫ e c x cos n ⁡ x d x = e c x cos n − 1 ⁡ x c 2 + n 2 ( c cos ⁡ x + n sin ⁡ x ) + n ( n − 1 ) c 2 + n 2 ∫ e c x cos n − 2 ⁡ x d x

Integrals involving the error function

∫ e c x ln ⁡ x d x = 1 c ( e c x ln ⁡ | x | − Ei ( c x ) ) (Ei is the exponential integral) ∫ x e c x 2 d x = 1 2 c e c x 2 ∫ e − c x 2 d x = π 4 c erf ⁡ ( c x ) (erf is the error function) ∫ x e − c x 2 d x = − 1 2 c e − c x 2 ∫ e − x 2 x 2 d x = − e − x 2 x − π e r f ( x ) ∫ 1 σ 2 π e − 1 2 ( x − μ σ ) 2 d x = 1 2 ( erf x − μ σ 2 )

Other integrals

∫ e x 2 d x = e x 2 ( ∑ j = 0 n − 1 c 2 j 1 x 2 j + 1 ) + ( 2 n − 1 ) c 2 n − 2 ∫ e x 2 x 2 n d x valid for any  n > 0 , where c 2 j = 1 ⋅ 3 ⋅ 5 ⋯ ( 2 j − 1 ) 2 j + 1 = ( 2 j ) ! j ! 2 2 j + 1   . (Note that the value of the expression is independent of the value of n, which is why it does not appear in the integral.) ∫ x x ⋅ ⋅ x ⏟ m d x = ∑ n = 0 m ( − 1 ) n ( n + 1 ) n − 1 n ! Γ ( n + 1 , − ln ⁡ x ) + ∑ n = m + 1 ∞ ( − 1 ) n a m n Γ ( n + 1 , − ln ⁡ x ) (for  x > 0 ) where a m n = { 1 if  n = 0 , 1 n ! if  m = 1 , 1 n ∑ j = 1 n j a m , n − j a m − 1 , j − 1 otherwise and Γ(x,y) is the gamma function ∫ 1 a e λ x + b d x = x b − 1 b λ ln ⁡ ( a e λ x + b ) when b ≠ 0 , λ ≠ 0 , and a e λ x + b > 0 . ∫ e 2 λ x a e λ x + b d x = 1 a 2 λ [ a e λ x + b − b ln ⁡ ( a e λ x + b ) ] when a ≠ 0 , λ ≠ 0 , and a e λ x + b > 0 .

Definite integrals

∫ 0 1 e x ⋅ ln ⁡ a + ( 1 − x ) ⋅ ln ⁡ b d x = ∫ 0 1 ( a b ) x ⋅ b d x = ∫ 0 1 a x ⋅ b 1 − x d x = a − b ln ⁡ a − ln ⁡ b for a > 0 ,   b > 0 ,   a ≠ b , which is the logarithmic mean ∫ 0 ∞ e − a x d x = 1 a ( Re ⁡ ( a ) > 0 ) ∫ 0 ∞ e − a x 2 d x = 1 2 π a ( a > 0 ) (the Gaussian integral) ∫ − ∞ ∞ e − a x 2 d x = π a ( a > 0 ) ∫ − ∞ ∞ e − a x 2 e − 2 b x d x = π a e b 2 a ( a > 0 ) (see Integral of a Gaussian function) ∫ − ∞ ∞ x e − a ( x − b ) 2 d x = b π a ( Re ⁡ ( a ) > 0 ) ∫ − ∞ ∞ x e − a x 2 + b x d x = π b 2 a 3 / 2 e b 2 4 a ( Re ⁡ ( a ) > 0 ) ∫ − ∞ ∞ x 2 e − a x 2 d x = 1 2 π a 3 ( a > 0 ) ∫ − ∞ ∞ x 2 e − a x 2 − b x d x = π ( 2 a + b 2 ) 4 a 5 / 2 e b 2 4 a ( Re ⁡ ( a ) > 0 ) ∫ − ∞ ∞ x 3 e − a x 2 + b x d x = π ( 6 a + b 2 ) b 8 a 7 / 2 e b 2 4 a ( Re ⁡ ( a ) > 0 ) ∫ 0 ∞ x n e − a x 2 d x = { Γ ( n + 1 2 ) 2 a n + 1 2 ( n > − 1 , a > 0 ) ( 2 k − 1 ) ! ! 2 k + 1 a k π a ( n = 2 k , k integer , a > 0 ) k ! 2 a k + 1 ( n = 2 k + 1 , k integer , a > 0 ) (!! is the double factorial) ∫ 0 ∞ x n e − a x d x = { Γ ( n + 1 ) a n + 1 ( n > − 1 , a > 0 ) n ! a n + 1 ( n = 0 , 1 , 2 , … , a > 0 ) ∫ 0 1 x n e − a x d x = n ! a n + 1 [ 1 − e − a ∑ i = 0 n a i i ! ] ∫ 0 ∞ e − a x b d x = 1 b   a − 1 b Γ ( 1 b ) ∫ 0 ∞ x n e − a x b d x = 1 b   a − n + 1 b Γ ( n + 1 b ) ∫ 0 ∞ e − a x sin ⁡ b x d x = b a 2 + b 2 ( a > 0 ) ∫ 0 ∞ e − a x cos ⁡ b x d x = a a 2 + b 2 ( a > 0 ) ∫ 0 ∞ x e − a x sin ⁡ b x d x = 2 a b ( a 2 + b 2 ) 2 ( a > 0 ) ∫ 0 ∞ x e − a x cos ⁡ b x d x = a 2 − b 2 ( a 2 + b 2 ) 2 ( a > 0 ) ∫ 0 2 π e x cos ⁡ θ d θ = 2 π I 0 ( x ) (I₀ is the modified Bessel function of the first kind) ∫ 0 2 π e x cos ⁡ θ + y sin ⁡ θ d θ = 2 π I 0 ( x 2 + y 2 )