Find ∫ 7 log⁡x.x dx

Category: QuestionsFind ∫ 7 log⁡x.x dx
Editor">Editor Staff asked 11 months ago

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
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

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