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Q.
if $\, g ( x ) = \int^x_0 \cos^4 \, t \, dt $ , then $g ( x + \pi ) $ equals
IIT JEEIIT JEE 1997Integrals
Solution:
Given, $ g(x)=\int\limits_{0}^{x} \cos ^{4} t d t$
$\Rightarrow g(x+\pi)=\int\limits_{0}^{\pi+x} \cos ^{4} t d t$
$=\int\limits_{0}^{\pi} \cos ^{4} t d t+\int\limits_{\pi}^{\pi+x} \cos ^{4} t d t$
$=I_{1}+I_{2}$
where, $ I_{1}=\int\limits_{0}^{\pi} \cos ^{4} t d t=g(\pi)$
and $ I_{2}=\int\limits_{\pi}^{\pi+x} \cos ^{4} t d t$
Put $ t=\pi+y$
$ \Rightarrow t =d y $
$I_{2} =\int\limits_{0}^{x} \cos ^{4}(y+\pi) d y $
$=\int\limits_{0}^{x}(-\cos y)^{4} d y=\int\limits_{0}^{x} \cos ^{4} y d y $
$=g(x) $
$ \therefore g(x+\pi) =g(\pi)+g(x)$