Question
The value of integration of log x / x2 is:
Solution
To evaluate the integral ∫ (logx / x2)​dx, we can use integration by parts. The formula for integration by parts is: ∫udv=uv−∫vdu We need to choose u and dv such that the integral ∫vdu is simpler than the original integral. Let's choose: u=logx ⟹ du= (1/x) ​dx dv = (1/x2) dx= x-2 dx ⟹ v=∫x-2 dx = {x-2+1 / (-2+1)} = x-1 / (-1) = -1/x Now, apply the integration by parts formula: ∫ (log x / x2)​ dx = (log x) (-1/x) - ∫(-1/x)(1/x) dx ∫ (log x / x2)​ dx = - log x / x + ∫ (1/x2) dx ∫ (log x / x2)​ dx = - log x / x – 1/x +C Therefore, the correct answer is option (B).
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