Test Index

AP EAPCET 04-Jul-2022 Shift 1 Paper

Section: Mathematics

© examsnet.com

Question : 76 of 160

Marks: +1, -0

\int\limits_{0}^{\pi} x(\sin^{2}(\sin x)+\cos^{2}(\cos x)) dx=

$\pi^{2}$
$\frac{\pi^{2}}{2}$
$2\pi$
$\frac{\pi}{4}$

Solution: 👈: Video Solution

Let

I=\int\limits_{0}^{\pi} x(\sin^{2}(\sin x)+\cos^{2}(\cos x)) dx

.....(i)

I=\int\limits_{0}^{\pi}(\pi-x)(\sin^{2}(\sin x)+\cos^{2}(\cos x)) dx

.....(ii)

Adding Eqs. (i) and (ii), we get

2I = \pi \int\limits_{0}^{\pi} (\sin^{2}(\sin x)+\cos^{2}(\cos x)) dx

2I = 2\pi \int\limits_{0}^{\frac{\pi}{2}} (\sin^{2}(\sin x)+\cos^{2}(\cos x)) dx.

I = \pi \int\limits_{0}^{\frac{\pi}{2}} \sin^{2}(\sin x) + \int\limits_{0}^{\frac{\pi}{2}} \cos^{2}(\cos x) dx

I = \pi \int\limits_{0}^{\frac{\pi}{2}} \sin^{2}(\cos x) + \int\limits_{0}^{\frac{\pi}{2}} \cos^{2}(\cos x) dx

= \pi \int\limits_{0}^{\frac{\pi}{2}} dx

I = \pi \times \frac{\pi}{2}

I = \frac{\pi^{2}}{2}

© examsnet.com

Go to Question:

Prev Question

Next Question