Find-the-reduction-formula-x-n-e-ax-dx-

Question Number 84101 by niroj last updated on 09/Mar/20

Find the reduction formula

\int x^{n} e^{ax} dx

Answered by MJS last updated on 09/Mar/20

\int x^{n} e^{a x} d x =

[t = a^{n + 1} x^{n + 1} \to d x = \frac{d t}{a^{n + 1} (n + 1) x^{n}}]

= \frac{1}{a^{n + 1} (n + 1)} \int e^{t^{\frac{1}{n + 1}}} d t =

this integral is the incomplete Gamma function

= \dots

= \frac{1}{{(- 1)}^{n} a^{n + 1}} Γ (n + 1, - a x) + C

Answered by TANMAY PANACEA last updated on 09/Mar/20

I_{n} = \int x^{n} e^{a x} d x

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

= \frac{x^{n} . e^{a x}}{a} - \frac{n}{a} I_{n - 1}

Commented by niroj last updated on 09/Mar/20

t h a n k t o a l l

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