Kathy Williams

2022-01-03

Find the indefinite integral.

$\int \frac{{\left(\mathrm{ln}x\right)}^{2}}{x}dx$

Pansdorfp6

Beginner2022-01-04Added 27 answers

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

$\Rightarrow \frac{1}{3}{\left(\mathrm{ln}\left|x\right|\right)}^{3}+c$

Given integral:

Substitute

Therefore, the given integral becomes

Step 2

We know that

Where, "c" is integration constant.

Substitute the value of

boronganfh

Beginner2022-01-05Added 33 answers

Given:

$\int \frac{{\mathrm{ln}}^{2}\left(x\right)}{x}dx$

Substitution$u=\mathrm{ln}\left(x\right)\Rightarrow \frac{du}{dx}=\frac{1}{x}$

$=\int {u}^{2}du$

$=\frac{{u}^{3}}{3}$

$=\frac{{\mathrm{ln}}^{3}\left(x\right)}{3}$

Answer:

$=\frac{{\mathrm{ln}}^{3}\left(x\right)}{3}+C$

Substitution

Answer:

Vasquez

Expert2022-01-07Added 669 answers

Step 1

Given:

Transform

Step 2

Evaluate

Substitute back

Add C

Step 3

Answer:

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