Question

Find the area inside integral f(x)=\(\frac{sec^4⁡(x)}{\sqrt{tan⁡(x)}}\) from x = 0 to π.

(a) π

(b) 0

(c) 1

(d) 2

This question was posed to me during a job interview.

This question is from Improper Integrals in section Integral Calculus of Engineering Mathematics

Answer 1

The correct answer is (b) 0

For explanation I would say: Given,F(x)=\(\int \frac{sec^4⁡ (x)}{\sqrt{tan⁡(x)}} dx\)

F(x)=\(\int \frac{sec^2⁡ (x) sec^2⁡ (x)}{\sqrt{tan⁡(x)}} dx\)

=\(\int \frac{1+t^2}{\sqrt{t}} dt\)

=\(\int [\frac{1}{\sqrt{t}}+t^{3/2}]dt\)

=\(2\sqrt{t}+\frac{2}{5} t^{5/2}\)

F(x)=\(\frac{2}{5} \sqrt{tan⁡(x)} [5+tan^2⁡(x)]\)

Now area inside a function f(x) from x=0 to π, is

F(π)-F(0)=0-0=0