Maths · Evaluation of definite integrals
\( \int_{100}^{2014} \frac{\sqrt{x}}{\sqrt{2114-x}+\sqrt{x}} d x= \)
- A. 1914
- B. 957
- C. 1007 \)
- D. \( \frac{2015}{2} \)
Step-by-step solution
Let I = ∫_{100}^{2014} √x/(√(2114-x)+√x) dx. Substitute x → 2114-x, then I = ∫_{100}^{2014} √(2114-x)/(√x+√(2114-x)) dx = ∫_{100}^{2014} (1 - √x/(√x+√(2114-x))) dx = 1914 - I, so 2I = 1914, I = 957.