Integrate 3 sec^2⁡x log⁡(tan⁡x) dx.

Category: QuestionsIntegrate 3 sec^2⁡x log⁡(tan⁡x) dx.
Editor">Editor Staff asked 11 months ago

Integrate 3 sec^2⁡x log⁡(tan⁡x) dx.
(a) -log⁡(tan⁡x) (tan⁡x-1)+C
(b) log⁡(tan⁡x) (sec⁡x+1)+C
(c) tan⁡x (log⁡(tan⁡x)-1)+C
(d) tan⁡x (log⁡sec⁡x +1)+C
I had been asked this question in my homework.
My question is based upon Integration by Parts topic in division Integrals of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

Right answer is (c) tan⁡x (log⁡(tan⁡x)-1)+C
Easiest explanation: By using∫ u.v dx=u∫ v dx-∫ u'(∫ v dx), we get
∫ log⁡(tan⁡x) sec^2x dx=log⁡(tan⁡x) ∫ sec^2 x⁡dx -∫ (log⁡tan⁡x)’∫ sec^2x dx
=tan⁡x log⁡(tan⁡x)-\(\int \frac{1}{tan⁡x} sec^2⁡x.tan⁡x \,dx\)
=tan x⁡log⁡(tan⁡x)-∫ sec^2⁡x dx
=tan x⁡log⁡(tan⁡x)-tan⁡x+C
=tan⁡x (log⁡(tan⁡x)-1)+C