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Find the value of `int_(1/2)^(2)e^(|x-1/x|)dx`. |
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Answer» Correct Answer - `esqrt(e)-1` Let `I=int_(1/2)^(2)e^(|x-1/x|)dx`…………….1 Put `x=1/t`, `:. I=-int_(0)^(1/2)e^(|t- 1/t|)((dt)/(t^(2)))=int_(1/2)^(2)e^(|x-1/x|)(dx)/(x^(2))`……………2 Adding 1 and 2 we get `2I=int_(1/2)^(2)e^(|x - 1/x|),(1+1/(x^(2)))dx` `=int_(1/2)^(1)e^(-(x-1/x)),(1+1/(x^(2)))dx+int_(1)^(2)e^((x-1/x)),(1+1/(x^(2)))dx` `=[-e^(-(x-1/x))]_(1//2)^(1)+[e^((x-1/x))]_(1)^(2)` `=-1+e^(3//2)+e^(3//2)-1` `=2(esqrt(e)-1)` `impliesI=esqrt(e)-1` |
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