We can solve the integral $\int\frac{\cos\left(\ln\left(x\right)\right)}{x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\ln\left(x\right)$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$
Replace $u$ with the value that we assigned to it in the beginning: $\ln\left(x\right)$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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