The value of the definite integral $\int\limits_{t+2 \pi}^{t+5 \pi / 2}\left\{\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)\right\} d x$ is equal to
Answer & explanation
Correct answer: option 3
Since $\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)$ is a periodic function with period $2 \pi$.
∴ $I=\int\limits_{t+2 \pi}^{t+5 \pi / 2}\left\{\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)\right\} d x$
$\Rightarrow I=\int\limits_0^{\pi / 2}\left\{\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)\right\} d x$
$\Rightarrow I=\int\limits_0^{\pi / 2} \frac{\pi}{2} d x=\frac{\pi^2}{4}$