Question

Find \(\int_1^{\sqrt{2}} (x^{\frac{8}{5}}-\frac{1}{x^{\frac{2}{5}}})^{\frac{5}{2}}dx\).

(a) -1

(b) 1

(c) 0

(d) ^1⁄5 – ^1⁄3 + ^1⁄1 – ^π⁄4

The question was asked during a job interview.

I need to ask this question from Integral Reduction Formula topic in portion Integral Calculus of Engineering Mathematics

Answer 1

The correct choice is (d) ^1⁄5 – ^1⁄3 + ^1⁄1 – ^π⁄4

The best I can explain: Simplifying we have

\(\int_1^{\sqrt{2}} (\frac{x^2-1}{x})^{\frac{5}{2}}dx\)

Substitute x=sec(t)

\(\int_{0}^{\frac{\pi}{4}} tan^6(t)dt\)

Now using the formula

\(\int_{0}^{\frac{\pi}{4}} tan^n(x)dx=\frac{1}{n-1}-\int_{0}^{\frac{\pi}{2}} tan^{n-2}(x)dx\)

We have\(=\frac{1}{5}-\frac{1}{3}+\frac{1}{1}-\frac{\pi}{4}\)