∫0(π)/2(sec⁡ x)/(nsec⁡ xn+cosec⁡ xn)dx=

Correct answer: (C) (π)/4. ∫0(π)/2(sec⁡ x)/(nsec⁡ xn+cos⁡ ec xn)dx ..... (i) ⇒ I=∫0(π)/2(sec⁡ π)/2-xnsec⁡ (π)/2-x+ cosce⁡(π)/2-xnndx ⇒ I=∫0(π)/2(cosce⁡ x)/(ncosce⁡xn + sec⁡ xn)dx ..... (ii) Adding equation (i) and (ii) 2I=∫0(π)/2dx ⇒ 2I=x0(π)/2 ⇒ (π)/4

Q80 / 150 · MHT CET 2016 · Mathematics

$\int_{0}^{\frac{π}{2}} (\frac{\sqrt[n]{\sec x}}{\sqrt[n]{\sec x} + \sqrt[n]{cosec x}}) d x =$

A
$\frac{π}{2}$
B
$\frac{π}{3}$
Correct answer
C
$\frac{π}{4}$
D
$\frac{π}{6}$

Solution

\int_{0}^{\frac{π}{2}} (\frac{\sqrt[n]{\sec x}}{\sqrt[n]{\sec x} + \sqrt[n]{\cos e c x}}) d x

..... (i)

\Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\sqrt[n]{\sec (\frac{π}{2} - x)}}{\sqrt[n]{\sec (\frac{π}{2} - x) + \sqrt[n]{cosce (\frac{π}{2} - x)}}} d x

\Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\sqrt[n]{cosce x}}{\sqrt[n]{cosce x} + \sqrt[n]{\sec x}} d x

..... (ii)
Adding equation (i) and (ii)

2 I = \int_{0}^{\frac{π}{2}} d x

\Rightarrow 2 I = {[x]}_{0}^{\frac{π}{2}}

\Rightarrow \frac{π}{4}

Hence, the correct answer is option: 'C'.

∫0π2sec⁡ xnsec⁡ xn+cosec⁡ xndx=

Solution

Relevant Questions

$\int_{0}^{\frac{π}{2}} (\frac{\sqrt[n]{\sec x}}{\sqrt[n]{\sec x} + \sqrt[n]{cosec x}}) d x =$