Find \int 6x(x^2+6)dx.

Category: QuestionsFind \int 6x(x^2+6)dx.
Editor">Editor Staff asked 11 months ago

Find \int 6x(x^2+6)dx.
(a) \frac{3x^4}{2}+18x^2+C
(b) \frac{3x^4}{2}-18x+C
(c) \frac{3x^4}{2}-18x^2+C
(d) \frac{3x^4}{2}+x^2+C
I had been asked this question in final exam.
My query is from Methods of Integration-1 in portion 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 choice is (a) \frac{3x^4}{2}+18x^2+C
To explain I would say: Let x^2=t
Differentiating w.r.t x, we get
2x dx=dt
\int 6x(x^2+6)dx=3\int (t+6) dt
3\int (t+6)dt=3\left (\frac{t^2}{2}+6t\right )=\frac{3t^2}{2}+18t
Replacing t with x^2
\int 6x(x^2+6)dx=\frac{3x^4}{2}+18x^2+C