Find \int_0^{π/4} \,2 \,tan⁡x \,dx.

Category: QuestionsFind \int_0^{π/4} \,2 \,tan⁡x \,dx.
Editor">Editor Staff asked 11 months ago

Find \int_0^{π/4} \,2 \,tan⁡x \,dx.
(a) log⁡2
(b) log⁡\sqrt{2}
(c) 2 log⁡2
(d) 0
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Query is from Fundamental Theorem of Calculus-1 in chapter Integrals of Mathematics – Class 12
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Editor">Editor Staff answered 11 months ago

The correct choice is (a) log⁡2
To explain: I=\int_0^{π/4} \,2 \,tan⁡x \,dx
F(x)=∫ 2 tan⁡x dx
=2∫ tan⁡x dx
=2 log⁡|sec⁡x|
Therefore, by using the fundamental theorem of calculus, we get
=2\left(log⁡|sec \frac{⁡π}{4}|-log⁡|sec⁡0|\right)=2 log⁡\sqrt{2}-log⁡1
=2 log⁡\sqrt{2}=log⁡(\sqrt{2})^2=log⁡2
I=\frac{8}{3} log⁡2-\frac{8}{3}-0+\frac{1}{3}=\frac{8}{3} log⁡2-\frac{7}{3}.