Use integration by parts to evaluate the integral . Note that evaluation may require integration by parts more than once.

Question

Use integration by parts to evaluate the integral  . Note that evaluation may require integration by parts more than once. ​

Quizplus · Accepted Answer

We can use integration by parts twice:
Let u = ln(x), dv = x^3 dx. Then, du = 1/x dx and v = x^4/4. 
Applying the formula,
∫ ln(x)x^3 dx = ln(x) x^4/4 - ∫ x^3 / 4x dx = ln(x)x^4/4 - ∫ x^2 / 4 dx

Let u = x^2/4, dv = dx. Then, du = x/2 dx and v = x. 
Applying the formula again,
∫ x^2 / 4 dx = x^3/12 - ∫ x/2 dx = x^3/12 - x^2/4 + C

Putting both results together,
∫ ln(x)x^3 dx = ln(x)x^4/4 - x^2 / 4 - x^3/12 + C = (3ln(2)-11)/12

Use Integration by Parts to Evaluate the Integral