1.

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)+CI had been asked this question in my homework.My question is based upon Integration by Parts topic in division Integrals of Mathematics – Class 12

Answer» 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


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