Question

What is the indefinite integral of `((lnx)^2)/x`?

Answer 1

`((ln x)^3)/3+C`

Explanation:

Do a substitution: `u=ln x`, `du = 1/x dx` to get:

`\int\ ((ln(x))^2)/x\ dx=\int\ u^{2}\ du=u^{3}/3+C=((ln x)^3)/3+C`

Answer 2

`=1/3 (ln x)^3 + C`

Explanation:

`int \((lnx)^2)/x \ dx`

another integration where knowing the commonest calculus patterns comes in very useful

so we have the pattern that `d/dx (f(x))^3 = 3 (f(x) )^2f'(x)`

and `d/dx ln (g(x)) = 1/(g(x)) g'(x)`

combining these ideas, watch what happens if we do

`d/dx (ln x)^3`

`= color{red}{3} (ln x)^2 * 1/x = color{red}{3} ((ln x)^2)/x`

so we do `color{red}{1/3} d/dx (ln x)^3 = d/dx (color{red}{1/3}(ln x)^3) = ((ln x)^2)/x`

if follows that

`int \ ((ln x)^2)/x \ dx =1/3 (ln x)^3 + C`