Question

Find the value of \(\int \frac{sec^4⁡(x)}{\sqrt{tan⁡(x)}} dx\).

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

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

(c) \(\frac{2}{5}\sqrt{tan⁡(x)}[6+tan^2⁡(x)]\)

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

The question was posed to me in an international level competition.

Question is from Improper Integrals in division Integral Calculus of Engineering Mathematics

Answer 1

Correct choice is (d) \(\frac{2}{5}\sqrt{tan⁡(x)}[5+tan^2⁡(x)]\)

Best explanation: Add constant automatically

Given, \(\int \frac{sec^4⁡(x)}{\sqrt{tan⁡(x)}} dx\)

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

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

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

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

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