Integration By Parts

To solve the integral

%5Cint%20x%5E3%20(%5Clog%20x)%5E2%20dx

, we use integration by parts. Let:

u%20%3D%20(%5Clog%20x)%5E2%20%5Cquad%20%5Ctext%7Band%7D%20%5Cquad%20dv%20%3D%20x%5E3%20dx

Then,

du%20%3D%20%5Cfrac%7B2%20%5Clog%20x%7D%7Bx%7D%20dx

and

v%20%3D%20%5Cfrac%7Bx%5E4%7D%7B4%7D

. Now apply the integration by parts formula:

%5Cint%20u%20dv%20%3D%20uv%20-%20%5Cint%20v%20du

Substitute the values of

u

du

v

, and

dv

, and integrate the resulting expression. After integrating, we obtain:

%5Cfrac%7B(%5Clog%20x)%5E2%20x%5E4%7D%7B4%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%5Cleft%5B%20(%5Clog%20x)%20x%5E4%20-%20%5Cfrac%7Bx%5E4%7D%7B16%7D%20%5Cright%5D%20%2B%20C

Thus, the correct answer is option (1).

Chapter Questions
1 MCQs