\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

 

Answers (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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