How do you use substitution to integrate $(\frac{ln (5 x)}{x}) \cdot d x$ $(ln(5x)/x)*dx$ ?

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Answer 1 · 2015-07-29T17:51:58

The integral is evaluated below.

We have to evaluate the integral,

$\int \frac{ln (5 x)}{x} \cdot d x$ $int Ln (5x)/x*dx$

Let, $ln (5 x) = t$ $Ln (5x) = t$
$\Rightarrow ln x + ln 5 = t$ $implies Ln x + Ln 5 = t$

$therefore 1/x + 0 = (dt)/dx$ (Differentiating with respect to $x$ )

$implies dt = dx/x$

Now, let us attack our problem and substitute $dt$ in place of $dx/x$ and the integral becomes,

$int t*dt = t^2/2 + C$

In terms of $x$ ,

$int Ln (5x)/x*dx = (Ln (5x))^2/2 + C$

How do you use substitution to integrate (ln(5x)/x)*dx(ln(5x)x)⋅dx(ln(5x)/x)*dx?