InterviewSolution
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InterviewSolution
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Evaluate:`int_1^2 1/((x+1)(x+2))dx`(ii) `int_1^2 1/(x(1+x^2))dx` |
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Answer» Let `(1)/(x(1+x^(2)))=(A)/(x)+(Bx+C)/((1+x^(2)))`. Then, `1-=A(1+x^(2))+(Bx+C)x`. Putting `x=0`, we get `A=1`. Comparing the coefficients of `x^(2)`, we get `A+B=0` or `B=-1`. Comparing coefficients f `x`, we get `C=0`. `:.(1)/(x(1+x^(2)))=[(1)/(x)-(x)/(1+x^(2))]` So, `int_(1)^(2)(dx)/(x(1+x^(2)))=int_(1)^(2)(dx)/(x)-(1)/(2)int_(1)^(2)(2x)/(1+x^(2))dx` `=[logx]_(1)^(2)-(1)/(2)[log(1+x^(2))]_(1)^(2)` `=[(3)/(2)(log2)-(1)/(2)(log5)]`. |
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