Question

\[ \int_0^{\frac{\pi}{2}} \frac{\sin^{100}x}{\sin^{100}x+\cos^{100}x}\,dx = \]

• A. \(\frac{\pi}{2}\)
• B. \(\frac{\pi}{4}\)
• C. \(100\)
• D. \(50\)

Hint:

For definite integrals from \(0\) to \(\frac{\pi}{2}\), use the transformation \(x\to\frac{\pi}{2}-x\).

Answer 1

The Correct Option is B

Concept: For integrals of the form: \[ I=\int_0^a f(x)\,dx \] we use the property: \[ I=\int_0^a f(a-x)\,dx \]

Step 1: Let: \[ I= \int_0^{\frac{\pi}{2}} \frac{\sin^{100}x}{\sin^{100}x+\cos^{100}x}\,dx \]

Step 2: Use the property: \[ I=\int_0^{\frac{\pi}{2}} f\left(\frac{\pi}{2}-x\right)\,dx \] Since: \[ \sin\left(\frac{\pi}{2}-x\right)=\cos x \] and \[ \cos\left(\frac{\pi}{2}-x\right)=\sin x \] So: \[ I= \int_0^{\frac{\pi}{2}} \frac{\cos^{100}x}{\cos^{100}x+\sin^{100}x}\,dx \]

Step 3: Add both expressions of \(I\). \[ 2I= \int_0^{\frac{\pi}{2}} \left[ \frac{\sin^{100}x}{\sin^{100}x+\cos^{100}x} + \frac{\cos^{100}x}{\sin^{100}x+\cos^{100}x} \right]dx \] \[ 2I= \int_0^{\frac{\pi}{2}}1\,dx \] \[ 2I=\frac{\pi}{2} \]

Step 4: Divide by \(2\). \[ I=\frac{\pi}{4} \] Therefore, \[ \boxed{\frac{\pi}{4}} \]