Question

Find \(\int_0^{π/4} \,2 \,tan⁡x \,dx\).

(a) log⁡2

(b) log⁡\(\sqrt{2}\)

(c) 2 log⁡2

(d) 0

The question was asked during an online interview.

Query is from Fundamental Theorem of Calculus-1 in chapter Integrals of Mathematics – Class 12

Answer 1

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

I=F(π/4)-F(0)

\(=2\left(log⁡|sec \frac{⁡π}{4}|-log⁡|sec⁡0|\right)=2 log⁡\sqrt{2}-log⁡1\)

\(=2 log⁡\sqrt{2}=log⁡(\sqrt{2})^2=log⁡2\)

I=\(\frac{8}{3} log⁡2-\frac{8}{3}-0+\frac{1}{3}=\frac{8}{3} log⁡2-\frac{7}{3}\).