How do I integrate by parts?

Lets use and example: y=x*sin(x). If we try to integrate this normally, we find it very hard. The integration by parts method splits up the function we are integrating into manageable bits. The formula we use to work this out is:


were the dU is the differential of U and dV is the differential of V. In our example y=x*sin(x), we let U=x and dV=sin(x). To find dU we differentiate x (dU=1). To find V we integrate dV (V=-cos(x)). Now we just stick everything in the formula:

Integral(xsin(x))= x*(-cos(x))-Integral((-cos(x))*1)=-x*cos(x)+sin(x) +C

where C is an arbitrary constant. This is the answer.

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