Is there any shortcut trick to evaluate $\int (xe^{x})^{n}dx$ where $n$ is a positive integer?

Question

Q)Is there any shortcut method to evaluate $$\int (xe^{x})^{n}dx,$$ where $n$ is a positive integer?

Ans) My approach:

Let $$I=\int (xe^{x})^{n}dx$$ Now let's solve by cases.

Case-1) When $n=1$, then we can write $$I=\int xe^{x}dx.$$ Now we have to integrate by applying Integration By Parts.

Therefore, $$I=x\int e^{x}dx-\int \frac{d}{dx}(x)\int e^{x}dx$$ $$\implies I= e^{x}(x-1)+C.$$

Case-2) When $n=2$, then we can write $$I=\int x^{2}e^{2x}dx.$$ Now again we have to integrate by applying Integration By Parts.

$$I=\int x^{2}e^{2x}dx$$ $$\implies I=x^{2}\int e^{2x}dx-\int\frac{d}{dx}(x^{2})\int e^{2x}dx$$ $$\implies I=\frac{x^{2}}{2}e^{2x}-\int xe^{2x}dx$$

Now after this step we have to integrate $$\int xe^{2x}dx$$ by applying Integration By Parts.

My doubt:

It will be very difficult to integrate $$\int (xe^{x})^{n} dx$$ by applying Integration By Parts for $n\geq 4$. The calculation will go on becoming lengthy. I want to know whether there is any shortcut method for this Integral ? Please help me out with this Integral.

Answer 1

$$I_n:=\int (x)^n e^{xn }dx $$ Let $xn =t $, $dx =\frac{dt}{n}$

$$I_n=\frac{1}{n^{n+1}}\int (t)^n e^{t }dx $$

Note that $\int e^t t dt = te^t -e^t = \sum\limits_{k=0}^{1}(-1)^{k} e^t t^ {1-k} P(1,k) $ such that $P(n,k )= \frac{n!}{k!}$

Assume that $I_{n-1} =\frac{1}{n^{n+1}}\sum\limits_{k=0}^{n-1}(-1)^{k} e^t t^ {n-1-k} P(n-1,k) $ $$I_{n}= \frac{1}{n^{n+1}} \bigg(e^t t^n - n\int t^{n-1}e^t dt \bigg)+C= \frac{1}{n^{n+1}}\bigg(e^t t^n-n\sum\limits_{k=0}^{n-1}(-1)^{k} e^t t^ {n-1-k} P(n-1,k)\bigg)$$ $$ =\frac{1}{n^{n+1}}\sum\limits_{k=0}^{n}(-1)^{k} e^t t^ {n-k} P(n,k) $$

Bu induction $I_{n}=\frac{1}{n^{n+1}}\sum\limits_{k=0}^{n}(-1)^{k} e^t t^ {n-k} P(n,k) $ for all $n$

$I_{n}= \frac{1}{n^{n+1}}\sum\limits_{k=0}^{n}(-1)^{k} e^t t^ {n-k} P(n,k) =\frac{1}{n^{n+1}}\sum\limits_{k=0}^{n}(-1)^{k} e^{ nx}{(nx)}^ {n-k} P(n,k)$

Answer 2

I record this here as it might be useful.

Let $-\infty<a<b\le\infty$ and $\lambda\in\mathbb C$ such that $b<\infty$ or $\Re\lambda>0$. Let $P$ be a polynomial. Then by integration by parts, \begin{align*} &&\int_a^bP(x)\,\mathrm e^{-\lambda x}\,\mathrm dx &=-\left[P(x)\,\frac{\mathrm e^{-\lambda x}}\lambda\right]_a^b+\frac1\lambda\int_a^bP'(x)\,\mathrm e^{-\lambda x}\,\mathrm dx, \end{align*} where $P'$ is again a polynomial. Applying the same with $P'(x)/\lambda$ in place of $P(x)$ yields $$\int_a^bP(x)\,\mathrm e^{-\lambda x}\,\mathrm dx=-\left[P(x)\,\frac{\mathrm e^{-\lambda x}}\lambda+P'(x)\,\frac{\mathrm e^{-\lambda x}}{\lambda^2}\right]_a^b+\frac1{\lambda^2}\int_a^bP''(x)\,\mathrm e^{-\lambda x}\,\mathrm dx.$$ Iterating, since eventually $P^{(k)}=0$ we can write $$\int_a^bP(x)\,\mathrm e^{-\lambda x}\,\mathrm dx=-\left[\sum_{k=0}^\infty P^{(k)}(x)\,\frac{\mathrm e^{-\lambda x}}{\lambda^{k+1}}\right]_a^b,$$ or in indefinite form, $$\int P(x)\,\mathrm e^{-\lambda x}\,\mathrm dx=-\sum_{k=0}^\infty\frac{P^{(k)}(x)\,\mathrm e^{-\lambda x}}{\lambda^{k+1}}+C.$$

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Apply with $\lambda=-n$ and $P(x)=x^n$. The $P^{(k)}(x)$ are easy to compute.

Answer 3

Note that

\begin{align} &\int (xe^{x})^{n}dx =\frac{d^n}{da^n} \bigg(\int e^{ax}dx\bigg)_{a=n} =\frac{d^n}{da^n}\left(\frac{e^{ax}}a\right)_{a=n}\\ \end{align}

Then, apply the product rule $$\frac{d^n}{da^n}f(a)g(a) =\sum_{k=0}^n \binom nk f^{(n-k)}(a) g^{(n)}(a) $$ to obtain

$$\int (xe^{x})^{n}dx= e^{nx} \sum_{k=0}^n \frac{(-1)^k n!}{n^{k+1}(n-k)!} {x^{n-k}} $$