Следващият списък съдържа интегралите на експоненциални функции. За интегралите на други функции, виж Таблични интеграли.
∫ e x d x = e x + C {\displaystyle \int e^{x}\,dx=e^{x}+C}
∫ a x d x = a x ln a + C {\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C} (в процеса на интегриране a x {\displaystyle a^{x}} се полага с e x ln a {\displaystyle e^{x\ln {a}}} )
∫ e a x d x = 1 a e a x + C {\displaystyle \int e^{ax}dx={\frac {1}{a}}e^{ax}+C}
∫ x e a x = 1 a e a x + i π 2 a 3 / 2 e r f ( i a x ) + C , e r f ( x ) = 2 x ∫ 0 x e − t 2 d t {\displaystyle \int {\sqrt {x}}e^{ax}={\frac {1}{a}}e^{ax}+{\frac {i{\sqrt {\pi }}}{2a^{3/2}}}erf(i{\sqrt {ax}})+C,\ erf(x)={\frac {2}{\sqrt {x}}}\int _{0}^{x}e^{-t^{2}}dt}
∫ x e x d x = ( x − 1 ) e x + C {\displaystyle \int xe^{x}dx=(x-1)e^{x}+C} [1]
∫ x e a x d x = ( x a − 1 a 2 ) e a x + C {\displaystyle \int xe^{ax}dx=\left({\frac {x}{a}}-{\frac {1}{a^{2}}}\right)e^{ax}+C} [2]
∫ 1 a + b e n x d x = 1 a n [ n x − ln a + b e n x ] + C {\displaystyle \int {\frac {1}{a+be^{nx}}}\ dx={\frac {1}{an}}[nx-\ln {a+be^{nx}}]+C}
∫ 1 1 + e x d x = ln e x 1 + e x = ln ( 1 + e x ) + C {\displaystyle \int {\frac {1}{1+e^{x}}}\ dx=\ln {\frac {e^{x}}{1+e^{x}}}=\ln {(1+e^{x})}+C}
∫ 1 a e n x + b e − n x d x = 1 n a b arctan e n x a b + C , a b > 0 {\displaystyle \int {\frac {1}{ae^{nx}+be^{-nx}}}\ dx={\frac {1}{n{\sqrt {ab}}}}\arctan {e^{nx}{\sqrt {\frac {a}{b}}}}+C,\ \ ab>0} [1]
= 1 2 m a b ln | b + e m x − a b b − e m x − a b | + C , a b < 0 {\displaystyle \ \ \ \ ={\frac {1}{2m{\sqrt {ab}}}}\ln {\left\vert {\frac {b+e^{mx}{\sqrt {-ab}}}{b-e^{mx}{\sqrt {-ab}}}}\right\vert }+C,\ \ ab<0}
∫ 1 a + b e n x = 1 n a ln a + b e n x − a a + b e n x + a + C , a > 0 {\displaystyle \int {\frac {1}{\sqrt {a+be^{nx}}}}={\frac {1}{n{\sqrt {a}}}}\ln {\frac {{\sqrt {a+be^{nx}}}-{\sqrt {a}}}{{\sqrt {a+be^{nx}}}+{\sqrt {a}}}}+C,\ \ a>0} [1]
= 2 n a arctan a + b e n x − a + C , a < 0 {\displaystyle \ \ \ \ ={\frac {2}{n{\sqrt {a}}}}\arctan {\frac {\sqrt {a+be^{nx}}}{\sqrt {-a}}}+C,\ \ a<0}
∫ exp ( a x ) d x = E i ( a x ) ln a + C {\displaystyle \int \exp {(a^{x})}\ dx={\frac {Ei(a^{x})}{\ln {a}}}+C}
∫ x e a x d x = e a x ( x a − 1 a 2 ) {\displaystyle \int xe^{ax}\ dx=e^{ax}\left({\frac {x}{a}}-{\frac {1}{a^{2}}}\right)}
∫ x 2 e x d x = e x ( x 2 − 2 x + 2 ) + C {\displaystyle \int x^{2}e^{x}\ dx=e^{x}(x^{2}-2x+2)+C}
∫ x 2 e a x d x = e a x ( x 2 a − 2 x a 2 + 2 a 3 ) + C {\displaystyle \int x^{2}e^{ax}\ dx=e^{ax}\left({\frac {x^{2}}{a}}-{\frac {2x}{a^{2}}}+{\frac {2}{a^{3}}}\right)+C}
∫ x 3 e a x d x = e a x ( x 3 a − 3 x 2 a 2 + 6 x a 3 − 6 a 4 ) + C {\displaystyle \int x^{3}e^{ax}\ dx=e^{ax}\left({\frac {x^{3}}{a}}-{\frac {3x^{2}}{a^{2}}}+{\frac {6x}{a^{3}}}-{\frac {6}{a^{4}}}\right)+C} [1]
∫ x 3 e x d x = e x ( x 3 − 3 x 2 + 6 x − 6 ) + C {\displaystyle \int x^{3}e^{x}\ dx=e^{x}(x^{3}-3x^{2}+6x-6)+C}
∫ x 4 e a x d x = e a x ( X 4 a − 4 x 3 a 2 + 12 x 2 a 3 − 24 x a 4 + 24 a 5 ) + C {\displaystyle \int x^{4}e^{ax}\ dx=e^{ax}\left({\frac {X^{4}}{a}}-{\frac {4x^{3}}{a^{2}}}+{\frac {12x^{2}}{a^{3}}}-{\frac {24x}{a^{4}}}+{\frac {24}{a^{5}}}\right)+C}
∫ x n e a x d x = x n e a x a − n a ∫ x n − 1 e a x d x , a ≠ 0 {\displaystyle \int x^{n}e^{ax}\ dx={\frac {x^{n}e^{ax}}{a}}-{\frac {n}{a}}\int x^{n-1}e^{ax}\ dx,\ \ a\neq 0} [1]
= ( − 1 ) n 1 a Γ [ 1 + n , − a x ] + C , Γ ( a , x ) = ∫ x ∞ t a − 1 e − t d t {\displaystyle \ \ \ \ =(-1)^{n}{\frac {1}{a}}\Gamma [1+n,-ax]+C,\ \ \Gamma (a,x)=\int _{x}^{\infty }t^{a-1}e^{-t}dt}
= e a x ( ∑ k = 0 n ( − 1 ) k k ! ( n k ) a k + 1 x n − k ) {\displaystyle \ \ \ \ =e^{ax}\left(\sum _{k=0}^{n}{\frac {(-1)^{k}k!{\binom {n}{k}}}{a^{k+1}}}x^{n-k}\right)}
∫ e a x 2 d x = − i π 2 a e r f ( i x a ) + C {\displaystyle \int e^{ax^{2}}\ dx=-i{\frac {\sqrt {\pi }}{2{\sqrt {a}}}}erf(ix{\sqrt {a}})+C}
∫ P n ( x ) e a x d x = e a x a ∑ k = 0 n ( − 1 ) k P ( k ) ( x ) a k + C {\displaystyle \int P_{n}(x)e^{ax}\ dx={\frac {e^{ax}}{a}}\sum {k=0}^{n}(-1)^{k}{\frac {P^{(k)}(x)}{a^{k}}}+C} [1], където Pn(x) е полином от n-та степен, а P(k)(x) е k-тата производна на многочлена.
(в процеса на интегриране 
се полага с 
)
[1]
[2]
![{\displaystyle \int {\frac {1}{a+be^{nx}}}\ dx={\frac {1}{an}}[nx-\ln {a+be^{nx}}]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2982f72eb957e6d4c6be2c2ad40d1b7847d6abf2)
[1]
[1]
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![{\displaystyle \ \ \ \ =(-1)^{n}{\frac {1}{a}}\Gamma [1+n,-ax]+C,\ \ \Gamma (a,x)=\int _{x}^{\infty }t^{a-1}e^{-t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06d32ba922f076685d9a6170f3be6d520c4f12a3)
[1], където Pn(x) е полином от n-та степен, а P(k)(x) е k-тата производна на многочлена.
