Question

Integration of x^n logx by parts

Answer 1

Answer:

What is the integration of x^n log x?

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Remember the expression,

ddxax=axlna

Now,

∫dxxnlnx

Instead, let’s consider the following function:

∫dxxnxt

Notice that if I take the derivative with respect to t it recovers our original integral at t=0

∂∂t∫dxxnxt∣∣∣t=0=∫dxxnlnx

Then, the integral of our new function is straightforward:

∫dxx(n+t)=1(n+t+1)x(n+t+1)

I just need to take the derivative of above function at t=0

∂∂t1(n+t+1)x(n+t+1)

=−x(n+t+1)(n+t+1)2+x(n+t+1)lnx(n+t+1)∣∣∣t=0

=−x(n+1)(n+1)2+x(n+1)lnx(n+1)+c

where c is just an arbitrary constant.