If $g(x)=\int_{0}^{x} \cos 4 t d t$, then $g(x+\pi)$ equals

JEE Main · 2012 · 29 Apr 2012 · Offline · Definite Integrals

g (x) = \int_{0}^{x} cos 4 t d t

, then

g (x + π)

equals

Answer: (B)

Here

g (x) = \int_{0}^{x} cos 4 t d t = \frac{sin 4 x}{4}

. Because

sin

has period

2 π

sin (4 x + 4 π) = sin 4 x

, so

g (x + π) = \frac{sin 4 ( x + π )}{4} = \frac{sin 4 x}{4} = g (x)

. Also

g (π) = \frac{sin 4 π}{4} = 0

, so

g (x + π) = g (x) = g (x) + g (π)

. The other choices fail:

g (x) / g (π)

is undefined,

2 g (x)

is not generally

g (x)

, and

g (x) \cdot g (π) = 0

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option

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