Prove that ∫√cos^n x dx/√cos^n x +√sin^n x from 0 to π/2=π/4.?

Question

511 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-12T17:47:26

Please refer to a Proof in the Explanation.

Prerequisite : #int_0^af(x)dx=int_0^af(a-x)dx#.

Let, #I=int_0^(pi/2){sqrt(cos^nx)/(sqrt(cos^nx)+sqrt(sin^nx))}dx......(1)#.

Then, using the above Result, we have,

#I=int_0^(pi/2){sqrt(cos^n(pi/2-x))/(sqrt(cos^n(pi/2-x))+sqrt(sin^n(pi/2-x))}dx#,

#:.I=int_0^(pi/2){(sqrt(sin^nx))/(sqrt(sin^nx)+sqrt(cos^nx))}dx......(2)#.

Adding #(1) and (2)#, we get,

#2I=int_0^(pi/2){(sqrt(cos^nx)+sqrt(sin^nx))/(sqrt(sin^nx)+sqrt(cos^nx))}dx, i.e.,#

#2I=int_0^(pi/2)1dx=[x]_0^(pi/2)=pi/2#.

# rArr I=pi/4#, as desired!

Feel and Spread the Joy of Maths.!