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}  

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Next page : One of the functions is sin(bx) or cos(bx) or a linear combination of them, the other is an exponential functionLast modified: May 20, 2018 @ 13:15