Integration By Parts Calculus Question

Question

Find the integral ∫(x^pi)(lnx) dx using integration by parts. Carefully explain all steps.

Answer 1

Integration by parts is best used when you have some part of the integrand that has an easy derivative that will simplify thge integral.

Integration by parts: Int (udv) = int(d(uv) - int (vdu) (This follows from product rule for differentials)

Int (udv) = uv - Int (vdu)

I use the acronym LItPET for choice of u (picture a stoned cat) logsm Inverse trig, power, exponential, and trig functions

Here lnx = u and x^πdx is dv du = 1/x dx and v = x^π+1/(π+1)

int (udv) = ln(x)*x^π+1/(π+1) - Int (x^π/(π+1))dx ( the integral term is x^π+1/(π+1)²

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