_ -3 2 ^ - 2 [(x+ )^3+ ^2(x+3 ) ] d x is equal to - Vidyadip

MCQ

$\int_{\frac{-3 \pi}{2}}^{\frac{-\pi}{2}}\left[(x+\pi)^3+\cos ^2(x+3 \pi)\right] d x$ is equal to

A
$\frac{\pi^4}{32}$
B
$\frac{\pi^4}{32}+\frac{\pi}{2}$
✓
$\frac{\pi}{2}$
D
$\frac{\pi}{4}-1$

✓

Answer

Correct option: C.

$\frac{\pi}{2}$

(C)
Let $I =\int_{\frac{-3 \pi}{2}}^{\frac{-\pi}{2}}\left[(x+\pi)^3+\cos ^2(x+3 \pi)\right] d x$
Put $x+\pi= t \Rightarrow d x= dt$
$\therefore \quad I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\left[t^3+\cos ^2(2 \pi+t)\right] d t$
$=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} t^3 d t+\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos ^2 t d t$
Here, $t^3$ is an odd function and $\cos ^2 t$ is an even function.
$\therefore \quad I =0+2 \int_0^{\frac{\pi}{2}} \cos ^2 tdt =2 \times \frac{\pi}{4}=\frac{\pi}{2}$

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