Example of Integration by Parts - Integration techniques

Illustration1:  Evaluate the following integral.

∫ xe^6x dx

Solution :

Thus, on some level, the difficulty here is the x that is in front of the exponential.  If that was not there we could do the integral.  Notice also, that in doing integration by parts anything that we wish for u will be differentiated.  Thus, it seems that choosing u = x will be a good choice as upon differentiating the x will drop out.

Here that we've selected u we know that dv will be everything else which remains.  Thus, here are the choices for u and dv also du and v.

u = x    dv = e^6x dx

du = dx           v = e^6x dx = 1/6e^6x

Then the integral is as follow:

∫ xe^6x dx = x/6 e^6x - ∫ 1/6 e^6x dx

= x/6 e^6x - 1/36 e^{6x + c}

Just once we have completed the last integral in the problem we will add in the constant of integration to obtain our final answer.