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$Evaluate \int_{- π}^{π} \frac{x \sin x d x}{e^{x} + 1}$

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a

π

b

2π

c

3π

d

4π

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detailed solution

Correct option is A

Let I=∫−ππ xsin⁡xdxex+1-----(1) Using property IV, we replace x by 0−x or −x∴ I=∫−ππ (−x)sin⁡(−x)dxe−x+1=∫−ππ exxsin⁡xdxex+1-----(2) Adding equations (1) and (2), we get 2I=∫−ππ ex+1 xsin⁡xdxex+1 2I=∫−ππxsinx dx=2∫0π xsin⁡xdx I=∫0π xsin⁡xdx I=∫0π (π−x)sin⁡(π−x)dx I=∫0π πsin⁡xdx−I 2I=π-cosx0π 2I=2π

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