# $\dpi{100} \int_{-3\pi /2}^{-\pi /2}\left [ (x+\pi )^{3}+cos^{2}(x+3\pi ) \right ]dx$    is equal to Option 1) $\frac{\pi ^{4}}{32}\;$ Option 2) $\; \; \frac{\pi ^{4}}{32}+\frac{\pi }{2}\;$ Option 3) $\; \frac{\pi }{2}\;$ Option 4) $\; \frac{\pi }{2}-1$

As learnt in concept

Properties of definite integration -

If $f\left ( x \right )$ is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

$\int_{-a}^{a}f(x)dx= 2\left \{ \int_{o}^{a} f(x)dx\right \}$

- wherein

Check even function $f(-x)=f(x)$ and symmetrical about y axis.

$I=\int_{\frac{-3\pi }{2}}^{-\frac{\pi }{2}}[(x+\pi)^{3}+cos^{2}(x+3\pi )]dx$

Put $x+\pi =t$

$I=\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}[t^{3}+cos^{2}t]dt$

$I=2\int_{0}^{\frac{\pi }{2}}cos^{2}t\:dt$

$=\int_{0}^{\frac{\pi }{2}}[1+Cos{2}t]dt$

=$[t]_{0}^{\frac{\pi }{2}}+\left [\frac{Sin2t}{2} \right ]_{0}^{\frac{\pi }{2}}$

=$\frac{\pi }{2}+0$

Option 1)

$\frac{\pi ^{4}}{32}\;$

This is incorrect option

Option 2)

$\; \; \frac{\pi ^{4}}{32}+\frac{\pi }{2}\;$

This is incorrect option

Option 3)

$\; \frac{\pi }{2}\;$

This correct option

Option 4)

$\; \frac{\pi }{2}-1$

This is incorrect option

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