Question

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

Answer 1

The integral is evaluated below.

Explanation:

We have to evaluate the integral,

`int Ln (5x)/x*dx`

Let, `Ln (5x) = 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`