Spisak integrala eksponencijalnih funkcija
Slijedi spisak integrala (antiderivacija funkcija) eksponencijalnih funkcija. Za potpuni spisak integrala funkcija, pogledati tabela integrala i spisak integrala. ∫ e c x d x = 1 c e c x {\displaystyle \int e^{cx}\;dx={\frac {1}{c}}e^{cx}} , ali ∫ e 2 x d x = 2 e 2 x {\displaystyle \int e^{2x}\;dx=2e^{2x}} ∫ a c x d x = 1 c ln ⁡ a a c x (za  a > 0 ,   a ≠ 1 ) {\displaystyle \int a^{cx}\;dx={\frac {1}{c\ln a}}a^{cx}\qquad {\mbox{(za }}a>0,{\mbox{ }}a\neq 1{\mbox{)}}} ∫ x e c x d x = e c x c 2 ( c x − 1 ) {\displaystyle \int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)} ∫ x 2 e c x d x = e c x ( x 2 c − 2 x c 2 + 2 c 3 ) {\displaystyle \int x^{2}e^{cx}\;dx=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)} ∫ x n e c x d x = 1 c x n e c x − n c ∫ x n − 1 e c x d x {\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx} ∫ e c x x d x = ln ⁡ | x | + ∑ i = 1 ∞ ( c x ) i i ⋅ i ! {\displaystyle \int {\frac {e^{cx}}{x}}\;dx=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}} ∫ 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 ) (za  n ≠ 1 ) {\displaystyle \int {\frac {e^{cx}}{x^{n}}}\;dx={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,dx\right)\qquad {\mbox{(za }}n\neq 1{\mbox{)}}} ∫ e c x ln ⁡ x d x = 1 c e c x ln ⁡ | x | − Ei ( c x ) {\displaystyle \int e^{cx}\ln x\;dx={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)} ∫ e c x sin ⁡ b x d x = e c x c 2 + b 2 ( c sin ⁡ b x − b cos ⁡ b x ) {\displaystyle \int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)} ∫ e c x cos ⁡ b x d x = e c x c 2 + b 2 ( c cos ⁡ b x + b sin ⁡ b x ) {\displaystyle \int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)} ∫ 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 {\displaystyle \int e^{cx}\sin ^{n}x\;dx={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;dx} ∫ 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 {\displaystyle \int e^{cx}\cos ^{n}x\;dx={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;dx} ∫ x e c x 2 d x = 1 2 c e c x 2 {\displaystyle \int xe^{cx^{2}}\;dx={\frac {1}{2c}}\;e^{cx^{2}}} ∫ e − c x 2 d x = π 4 c erf ( c x ) {\displaystyle \int e^{-cx^{2}}\;dx={\sqrt {\frac {\pi }{4c}}}{\mbox{erf}}({\sqrt {c}}x)} ( erf {\displaystyle {\mbox{erf}}} je funkcija greške (error function)) ∫ x e − c x 2 d x = − 1 2 c e − c x 2 {\displaystyle \int xe^{-cx^{2}}\;dx=-{\frac {1}{2c}}e^{-cx^{2}}} ∫ 1 σ 2 π e − ( x − μ ) 2 / 2 σ 2 d x = 1 2 ( 1 + erf x − μ σ 2 ) {\displaystyle \int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2}}(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}})} ∫ 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 valjano za  n > 0 , {\displaystyle \int e^{x^{2}}\,dx=e^{x^{2}}\left(\sum _{j=0}^{n-1}c_{2j}\,{\frac {1}{x^{2j+1}}}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}}}{x^{2n}}}\;dx\quad {\mbox{valjano za }}n>0,} pri čemu je c 2 j = 1 ⋅ 3 ⋅ 5 ⋯ ( 2 j − 1 ) 2 j + 1 = ( 2 j ) ! j ! 2 2 j + 1   . {\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {(2j)\,!}{j!\,2^{2j+1}}}\ .}