One function is in the form x to the power of n and the other can be integrated at least n times -

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Here we can set u= xⁿ, then after one partial integration we get an integrand with a factor xⁿ. We may then repeat the process until we get to 1. The example on the previous page is a trivial example is an example of this, so let us look at a somewhat more complex example.

$\int {{x^2}\sin (x)dx}$

$\left| {\begin{array}{*{20}{c}} {u = {x^2}} \\ {\frac{{dv}}{{dx}} = \sin x} \\ {\frac{{du}}{{dx}} = 2x} \\ {v = - \cos x} \end{array}} \right.$

$\begin{gathered} \int {{x^2}\sin (x)dx} = - {x^2}\cos x - \int {2x( - \cos x)} \;dx \hfill \\ \quad = - {x^2}\cos x + 2\int {x\cos x\;dx} \hfill \\ \end{gathered}$

We then do an integration by parts again.

$\left| {\begin{array}{*{20}{c}} {u = x} \\ {\frac{{dv}}{{dx}} = \cos x} \\ {\frac{{du}}{{dx}} = 1} \\ {v = \sin x} \end{array}} \right.$

$\begin{gathered} - {x^2}\cos x + 2\int {x\cos x\;dx} = \hfill \\ \quad = - {x^2}\cos x + 2(x\sin x - \int {1\sin x\,dx} ) \hfill \\ \quad = - {x^2}\cos x + 2(x\sin x + \cos x) + C \hfill \\ \quad = - {x^2}\cos x + 2x\sin x + 2\cos x + C \hfill \\ \end{gathered}$

Up a level : Integrals
Previous page : Integration by parts
Next page : One of the functions is sin(bx) or cos(bx) or a linear combination of them, the other is an exponential function

Last modified: May 20, 2018 @ 13:15