QJEEM12Ma71d3: JEE Main 2012 Mathematics - Class 12 JEEM12Ma71d3 Solution: If then equals

Question

Verified Solution • 2025–26 Session moderate

If

g\left( x \right) = \int\limits_0^x {\cos 4t\,dt,}

then

g\left( {x + \pi } \right)

equals

Accepted Answer

\left( {b,c} \right)g\left( {x + \pi } \right) = \int\limits_0^{x + \pi } {\cos \,4t\,dt}

= \int\limits_0^\pi {\cos 4tdt + \int\limits_\pi ^{\pi + x} {\cos 4t\,dt} }

= g\left( \pi \right) + \int\limits_0^x {\cos \,4t\,dt}

Putting

t = \pi + y

in second integral, we get

\int\limits_x^{\pi + x} {\cos \,4t\,dt = \int\limits_0^\pi {\cos \,4t\,dt} }

= g\left( \pi \right) + g\left( x \right) = g\left( x \right) - g\left( \pi \right)

\therefore

\left. {g\left( \pi \right) = 0} \right)

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