1. Tardigrade
  2. Question
  3. Mathematics
  4. The value of definite integral ∫ limits-ππ (x2/1+ sin x+√1+ sin 2 x) d x, is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The value of definite integral $\int\limits_{-\pi}^\pi \frac{x^2}{1+\sin x+\sqrt{1+\sin ^2 x}} d x$, is

Integrals

Solution:

image
$\therefore 2 I=\int\limits_{-\pi}^\pi \frac{x^2\left(2+2 \sqrt{1+\sin ^2 x}\right)}{\left(1+\sqrt{1+\sin ^2 x}\right)^2-\sin ^2 x} d x$
So, $I=\int\limits_{-\pi}^\pi \frac{x^2\left(1+\sqrt{1+\sin ^2 x}\right)}{2\left(1+\sqrt{1+\sin ^2 x}\right)} d x=\frac{1}{2} \int\limits_{-\pi}^\pi x^2 d x=\frac{\pi^3}{3}$.