Question

Find ∫ 7 log⁡x.x dx

(a) \(\frac{7}{2} (log⁡x-x)+C\)

(b) –\(\frac{7}{2} (x^2 log⁡x-x^3)+C\)

(c) \(\frac{7}{2} (x^2 log⁡x-x)+C\)

(d) (x^2 log⁡x+x)+C

The question was asked in a job interview.

The origin of the question is Integration by Parts topic in section Integrals of Mathematics – Class 12

Answer 1

Right option is (c) \(\frac{7}{2} (x^2 log⁡x-x)+C\)

To explain I would say: ∫ 7 log⁡x.x dx=7∫ log⁡x.x dx

Using ∫ u.v dx=u∫ v dx-∫ u'(∫ v dx) , we get

7∫ log⁡x.x dx=7(log⁡x ∫ x dx-(log⁡x)’∫ x dx)

=\(7\left (\frac{x^2 log⁡x}{2}-\frac{1}{x}.\frac{x^2}{2}\right)\)

=\(\frac{7}{2} (x^2 log⁡x-x)+C\)