Primitives de fonctions exponentielles

Cet article donne les primitives de fonctions exponentielles.

On suppose a≠0.

$\int \mathrm {e} ^{ax+b}\,\mathrm {d} x={\frac {1}{a}}\mathrm {e} ^{ax+b}+C$$\int \mathrm {e} ^{ax+b}\,\mathrm {d} x={\frac {1}{a}}\mathrm {e} ^{ax+b}+C$

$\int {\frac {\mathrm {e} ^{ax}}{x}}\,\mathrm {d} x=\ln |x|+\sum _{n=1}^{+\infty }{\frac {(ax)^{n}}{nn!}}+C$$\int {\frac {\mathrm {e} ^{ax}}{x}}\,\mathrm {d} x=\ln |x|+\sum _{n=1}^{+\infty }{\frac {(ax)^{n}}{nn!}}+C$

$\int x^{n}\mathrm {e} ^{ax}\,\mathrm {d} x={\frac {1}{a}}x^{n}\mathrm {e} ^{ax}-{\frac {n}{a}}\int x^{n-1}\mathrm {e} ^{ax}\,\mathrm {d} x$$\int x^{n}\mathrm {e} ^{ax}\,\mathrm {d} x={\frac {1}{a}}x^{n}\mathrm {e} ^{ax}-{\frac {n}{a}}\int x^{n-1}\mathrm {e} ^{ax}\,\mathrm {d} x$ (n ∈ ℤ\{-1})

Soit P une fonction polynôme. On note P’ sa dérivée.

$\int P(x)\mathrm {e} ^{ax}\,\mathrm {d} x={\frac {1}{a}}P(x)\mathrm {e} ^{ax}-{\frac {1}{a}}\int P^{\prime }(x)\mathrm {e} ^{ax}\,\mathrm {d} x$$\int P(x)\mathrm {e} ^{ax}\,\mathrm {d} x={\frac {1}{a}}P(x)\mathrm {e} ^{ax}-{\frac {1}{a}}\int P^{\prime }(x)\mathrm {e} ^{ax}\,\mathrm {d} x$

$\int \mathrm {e} ^{ax}\ln x\,\mathrm {d} x={\frac {1}{a}}\mathrm {e} ^{ax}\ln |x|-{\frac {1}{a}}\int {\frac {\mathrm {e} ^{ax}}{x}}\,\mathrm {d} x$$\int \mathrm {e} ^{ax}\ln x\,\mathrm {d} x={\frac {1}{a}}\mathrm {e} ^{ax}\ln |x|-{\frac {1}{a}}\int {\frac {\mathrm {e} ^{ax}}{x}}\,\mathrm {d} x$