d x is equal to :

Question:

$\int_{-\pi}^{\pi}|\pi-| x|| d x$ is equal to :

 

  1. (1) $\sqrt{2} \pi^{2}$

  2. (2) $2 \pi^{2}$

  3. (3) $\pi^{2}$

  4. (4) $\frac{\pi^{2}}{2}$


Correct Option: , 3

Solution:

$I=\int_{-\pi}^{\pi}|\pi-| x|| d x \quad[\because|\pi-| x||$ is even $]$

$=2 \int_{0}^{\pi}|\pi-| x|| d x$

$=2 \int_{0}^{\pi}(\pi-x) d x$

$=2\left[\pi x-\frac{x^{2}}{2}\right]_{0}^{\pi}=2\left(\pi^{2}-\frac{\pi^{2}}{2}\right)=\pi^{2}$

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