The Gateway to Computer Science Excellence
+2 votes
82 views

The integral $$\int _0^{\frac{\pi}{2}} \frac{\sin^{50} x}{\sin^{50}x +\cos^{50}x} dx$$ equals

  1. $\frac{3 \pi}{4}$
  2. $\frac{\pi}{3}$
  3. $\frac{\pi}{4}$
  4. none of these
in Calculus by Veteran (431k points)
edited by | 82 views

1 Answer

+4 votes
Best answer
Let $\displaystyle I=\int_{0}^{\frac{\pi}{2}} \frac{\sin^{50}x}{\sin^{50}x+\cos^{50}x}dx\qquad \to (1)$

By using property of definite integration

$\displaystyle \int_{0}^{a}f(x)dx=\int_{0}^{a}f(a-x)dx$

$\displaystyle I=\int_{0}^{\frac{\pi}{2}} \frac{\cos^{50}x}{\cos^{50}x+\sin^{50}x}dx\qquad \to (2)$

Adding $(1)$ and $(2),$

$\displaystyle 2I=\int_{0}^{\frac{\pi}{2}} \frac{\sin^{50}x+\cos^{50}x}{\sin^{50}x+\cos^{50}x}dx$

$\displaystyle \implies 2I=\int_{0}^{\frac{\pi}{2}}1\;dx$

$\displaystyle \implies 2I=\left[x\right]_{0}^{\frac{\pi}{2}}$

$\implies 2I=\left[ \frac{\pi}{2}\right]$

$\implies I=\frac{\pi}{4}$

So, answer is (C).
by Boss (12.9k points)
selected by
Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,737 questions
57,291 answers
198,209 comments
104,888 users