"Find the indefinite integral. ∫(ln⁡x)2xdx" solved and explained

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Kathy Williams

Answered question

2022-01-03

Find the indefinite integral.

\int \frac{{(\ln x)}^{2}}{x} d x

Answer & Explanation

Pansdorfp6

Beginner2022-01-04Added 27 answers

Step 1
Given integral:

\int \frac{{(\ln x)}^{2}}{x} d x

Substitute

t = \ln (x)

and differentiate it w.r.t x

\frac{d t}{d x} = \frac{d}{d x} \ln (x)

\frac{d t}{d x} = \frac{1}{x}

d t = \frac{1}{x} d x

Therefore, the given integral becomes

\int t^{2} d t

Step 2
We know that

\int x^{n} d = \frac{x^{n + 1}}{n + 1} + c

Where, "c" is integration constant.

\int t^{2} d t = \frac{1}{3} t^{3} + c

Substitute the value of

t = \ln (x)

in above equation, we get

\Rightarrow \frac{1}{3} {(\ln | x |)}^{3} + c

boronganfh

Beginner2022-01-05Added 33 answers

Given:

\int \frac{\ln^{2} (x)}{x} d x

Substitution

u = \ln (x) \Rightarrow \frac{d u}{d x} = \frac{1}{x}

= \int u^{2} d u

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

= \frac{\ln^{3} (x)}{3}

Answer:

= \frac{\ln^{3} (x)}{3} + C

Vasquez

Expert2022-01-07Added 669 answers

Step 1
Given:
$\int \frac{\ln (x)^{2}}{x} d x$
Transform
$\int t^{2} d t$
Step 2
Evaluate
$\frac{t^{3}}{3}$
Substitute back
$\frac{\ln (x)^{3}}{3}$
Add C
Step 3
Answer:
$\frac{\ln (x)^{3}}{3} + C$

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