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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)
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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).
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