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Evaluate:`int_(-pi)^pi(xsinx dx)/(e^x+1)` |
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Answer» Let `I=pi_(-pi)^(pi)(x sin x dx)/(e^(x)+1)`……………..1 Replacing `x` by `0-x` or `r-x`, we get `I=int_(-pi)^(pi)((-x)sin(-x)dx)/(e^(-x)+1)=int_(-pi)^(pi)(e^(x)xsin xdx)/(e^(x)+1)`…………….2 Adding 1 and 2 we get `2I=int_(-pi)^(pi) x sin x dx` or `I=int_(0)^(pi)xsinxdx` `int_(0)^(pi)(pi-x)sin(pi-x)dx=int_(0)^(pi)x sin x dx-I` or `I=pi` |
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