One Problem at a Time

Integration by Parts (Reduction Formula)

Problem 237: Show that

(1)   \begin{equation*} \int x^n \cos x \ dx = x^n \sin x - n \int x^{n-1} \sin x \ 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}dx,\end{equation*}

and

(3)   \begin{equation*} dv = \cos x \ dx \implies v = \sin x.\end{equation*}

By the integration by parts formula,

(4)   \begin{align*} \int x^n \cos x \ dx & = x^n \sin x - \int \sin x \cdot n \cdot x^{n-1} \ dx \\ & = x^n \sin x - n \int x^{n-1}\sin x \ dx. \end{align*}

This shows what we wanted to show.

(5)   \begin{equation*} \int x^n \cos x \ dx = x^n \sin x - n \int x^{n-1} \sin x \ dx\end{equation*}

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