Question

Evaluate : $\int\limits^3_1 {\frac{\cos (\log x)}{x}} \, dx$

Answer 1

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Answer 2

we have to integrate the $\int\limits^3_1{\frac{cos(logx)}{x}}\,dx$

Let logx = p.......(1)

differentiating both sides,

we get, dx/x = dp........(2)

now lower limit , p = log(1) = 0

upper limit, p = log(3) = log3

putting equations (1) and (2) in above expression.

we get, $\int\limits^{log3}_0{cos(p)}\,dp$

= $[sinp]^{log3}_0$

= $sin(log3)-sin0$

= sin(log3)

hence, $\int\limits^3_1{\frac{cos(logx)}{x}}\,dx=sin(log3)$