One Problem at a Time

Integration by Parts (Reduction Formula)

Problem 238: Show that

(1)   \begin{equation*} \int x^n e^{ax} \ dx = \frac{x^ne^{ax}}{a} - \frac{n}{a}\int x^{n-1}e^{ax} \ dx.\end{equation*}

Solution: One can use integration by parts to show this. That is, let

(2)   \begin{equation*} u = x^n \implies du = nx^{n-1},\end{equation*}

and

(3)   \begin{equation*} dv = e^{ax}dx \implies v = \frac{1}{a}e^{ax}, a \ne 0. \end{equation*}

Using the integration by parts formula,

(4)   \begin{align*} \int x^n e^{ax} \ dx & = x^n \cdot \frac{1}{a}e^{ax} - \int \frac{1}{a}e^{ax} \cdot nx^{n-1}dx \\ & = \frac{x^ne^{ax}}{a} - \frac{n}{a} \int x^{n-1}e^{ax} \ dx, a \ne 0. \end{align*}

This completes the proof.

(5)   \begin{equation*} \int x^n e^{ax} \ dx = \frac{x^ne^{ax}}{a} - \frac{n}{a}\int x^{n-1}e^{ax} \ dx, a\ne 0.\end{equation*}

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