Kathy Williams

2022-01-03

Find the indefinite integral.
$\int \frac{{\left(\mathrm{ln}x\right)}^{2}}{x}dx$

Pansdorfp6

Step 1
Given integral:
$\int \frac{{\left(\mathrm{ln}x\right)}^{2}}{x}dx$
Substitute $t=\mathrm{ln}\left(x\right)$ and differentiate it w.r.t x
$\frac{dt}{dx}=\frac{d}{dx}\mathrm{ln}\left(x\right)$
$\frac{dt}{dx}=\frac{1}{x}$
$dt=\frac{1}{x}dx$
Therefore, the given integral becomes
$\int {t}^{2}dt$
Step 2
We know that $\int {x}^{n}d=\frac{{x}^{n+1}}{n+1}+c$
Where, "c" is integration constant.
$\int {t}^{2}dt=\frac{1}{3}{t}^{3}+c$
Substitute the value of $t=\mathrm{ln}\left(x\right)$ in above equation, we get
$⇒\frac{1}{3}{\left(\mathrm{ln}|x|\right)}^{3}+c$

boronganfh

Given:
$\int \frac{{\mathrm{ln}}^{2}\left(x\right)}{x}dx$
Substitution $u=\mathrm{ln}\left(x\right)⇒\frac{du}{dx}=\frac{1}{x}$
$=\int {u}^{2}du$
$=\frac{{u}^{3}}{3}$
$=\frac{{\mathrm{ln}}^{3}\left(x\right)}{3}$
$=\frac{{\mathrm{ln}}^{3}\left(x\right)}{3}+C$

Vasquez

Step 1
Given:
$\int \frac{\mathrm{ln}\left(x{\right)}^{2}}{x}dx$
Transform
$\int {t}^{2}dt$
Step 2
Evaluate
$\frac{{t}^{3}}{3}$
Substitute back
$\frac{\mathrm{ln}\left(x{\right)}^{3}}{3}$
Step 3
$\frac{\mathrm{ln}\left(x{\right)}^{3}}{3}+C$