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NDA 2024 Maths · Mathematics · Definite Integrals · hard

The value of ∫0^π/2(sin x)/(sin x+cos x) dx is

Single correct option

The value of ∫0π/2sin⁡xsin⁡x+cos⁡x dx\displaystyle\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}\,dx is
  1. A. π2\frac{\pi}{2}
  2. B. π4\frac{\pi}{4}
  3. C. 11
  4. D. π8\frac{\pi}{8}
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Answer: (B) π4\frac{\pi}{4}

Let I=∫0π/2sin⁡xsin⁡x+cos⁡xdxI=\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}dx. Using x→π2−xx\to\frac\pi2-x, I=∫0π/2cos⁡xsin⁡x+cos⁡xdxI=\int_0^{\pi/2}\frac{\cos x}{\sin x+\cos x}dx. Adding, 2I=∫0π/21 dx=π22I=\int_0^{\pi/2}1\,dx=\frac\pi2, so I=π4I=\frac\pi4. Trap: π2\frac\pi2 is the value of 2I2I and not II.
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