Let g (x) = sin x + cos x and f (x) = ….

Q: Let g (x) = sin x + cos x and $ \large f(x) = \left\{\begin{array}{ll} \frac{|x|}{x} \; , x \ne 0 \\ 0 \; , x = 0 \end{array} \right. $ then value of $\large \int_{-\pi/4}^{2\pi}f(g(x))dx $ is equal to

(A) π/2

(B) π/4

(C) π

(D) none of these

Sol: $ \large f(g(x)) = \left\{\begin{array}{lll} 1 \; ,  sinx + cosx  > 0 \\ -1 \; , sinx + cosx  < 0  \\ 0 \; , sinx + cosx  = 0\end{array} \right. $

Now sin x + cos x > 0

$\large \sqrt{2}sin(x+\frac{\pi}{4}) > 0 $

$\large \int_{-\pi/4}^{2\pi}f(g(x))dx $

$\large = \int_{-\pi/4}^{3\pi/4}1 dx + \int_{3\pi/4}^{7\pi/4}-1 dx + \int_{7\pi/4}^{2\pi}1 dx $

$\large = \pi – \pi + \frac{\pi}{4} = \frac{\pi}{4} $

Hence (B) is the correct answer.