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Find \(\int_0^{π/4} \,2 \,tan⁡x \,dx\).(a) log⁡2(b) log⁡\(\sqrt{2}\)(c) 2 log⁡2(d) 0The question was asked during an online interview.Query is from Fundamental Theorem of Calculus-1 in chapter Integrals of Mathematics – Class 12

Answer»

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


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