Integrate ((ln(x))/x)^2

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

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Answer

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxcosh\left(x\right)+\int cosh\left(x\right)\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxdx-\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$

Step by step solution

Problem

$\int\left(\frac{\ln\left(x\right)}{x}\right)^2dx$
1

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

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

Rewrite the fraction $\frac{\ln\left(x\right)^2}{x^2}$ inside the integral as a product $\frac{1}{x^2}\ln\left(x\right)^2$

$\int\frac{1}{x^2}\ln\left(x\right)^2dx$
3

Use the integration by parts theorem to calculate the integral $\int\frac{1}{x^2}\ln\left(x\right)^2dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
4

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\frac{1}{x^2}}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
5

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\ln\left(x\right)^2dx}\\ \displaystyle{\int dv=\int \ln\left(x\right)^2dx}\end{matrix}$
6

Solve the integral

$v=\int\ln\left(x\right)^2dx$
7

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx-\int cosh\left(x\right)\int\ln\left(x\right)^2dxdx$
8

Use the integration by parts theorem to calculate the integral $\int\int\ln\left(x\right)^2dxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
9

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
10

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\ln\left(x\right)^2dxdx}\\ \displaystyle{\int dv=\int \int\ln\left(x\right)^2dxdx}\end{matrix}$
11

Solve the integral

$v=\int\int\ln\left(x\right)^2dxdx$
12

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx-\left(\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\ln\left(x\right)^2dxdxdx\right)$
13

Multiply $\left(\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)+-\int cosh\left(x\right)\int\int\ln\left(x\right)^2dxdxdx\right)$ by $-1$

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int cosh\left(x\right)\int\int\ln\left(x\right)^2dxdxdx-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
14

Use the integration by parts theorem to calculate the integral $\int\int\int\ln\left(x\right)^2dxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
15

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
16

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\ln\left(x\right)^2dxdxdx}\\ \displaystyle{\int dv=\int \int\int\ln\left(x\right)^2dxdxdx}\end{matrix}$
17

Solve the integral

$v=\int\int\int\ln\left(x\right)^2dxdxdx$
18

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\ln\left(x\right)^2dxdxdxdx-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
19

Use the integration by parts theorem to calculate the integral $\int\int\int\int\ln\left(x\right)^2dxdxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
20

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
21

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\int\ln\left(x\right)^2dxdxdxdx}\\ \displaystyle{\int dv=\int \int\int\int\ln\left(x\right)^2dxdxdxdx}\end{matrix}$
22

Solve the integral

$v=\int\int\int\int\ln\left(x\right)^2dxdxdxdx$
23

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)-\left(\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\int\ln\left(x\right)^2dxdxdxdxdx\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
24

Multiply $\left(\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)+-\int cosh\left(x\right)\int\int\int\int\ln\left(x\right)^2dxdxdxdxdx\right)$ by $-1$

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int cosh\left(x\right)\int\int\int\int\ln\left(x\right)^2dxdxdxdxdx-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
25

Use the integration by parts theorem to calculate the integral $\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
26

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
27

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\int\int\ln\left(x\right)^2dxdxdxdxdx}\\ \displaystyle{\int dv=\int \int\int\int\int\ln\left(x\right)^2dxdxdxdxdx}\end{matrix}$
28

Solve the integral

$v=\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdx$
29

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdx-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
30

Use the integration by parts theorem to calculate the integral $\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
31

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
32

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdx}\\ \displaystyle{\int dv=\int \int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdx}\end{matrix}$
33

Solve the integral

$v=\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdx$
34

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)-\left(\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdx\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
35

Multiply $\left(\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)+-\int cosh\left(x\right)\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdx\right)$ by $-1$

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int cosh\left(x\right)\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdx-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
36

Use the integration by parts theorem to calculate the integral $\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
37

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
38

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdx}\\ \displaystyle{\int dv=\int \int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdx}\end{matrix}$
39

Solve the integral

$v=\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdx$
40

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdx-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
41

Use the integration by parts theorem to calculate the integral $\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
42

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
43

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdx}\\ \displaystyle{\int dv=\int \int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdx}\end{matrix}$
44

Solve the integral

$v=\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdx$
45

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)-\left(\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdx\right)-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
46

Multiply $\left(\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)+-\int cosh\left(x\right)\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdx\right)$ by $-1$

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)+\int cosh\left(x\right)\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdx-\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
47

Use the integration by parts theorem to calculate the integral $\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
48

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
49

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdx}\\ \displaystyle{\int dv=\int \int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdx}\end{matrix}$
50

Solve the integral

$v=\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdx$
51

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdx-\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
52

Use the integration by parts theorem to calculate the integral $\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdu$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
53

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=cosh\left(x\right)}\\ \displaystyle{du=cosh\left(x\right)dx}\end{matrix}$
54

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdx}\\ \displaystyle{\int dv=\int \int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdx}\end{matrix}$
55

Solve the integral

$v=\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdx$
56

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxcosh\left(x\right)-\left(\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxcosh\left(x\right)-\int cosh\left(x\right)\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxdx\right)-\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$
57

Multiply $\left(\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxcosh\left(x\right)+-\int cosh\left(x\right)\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxdx\right)$ by $-1$

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxcosh\left(x\right)+\int cosh\left(x\right)\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxdx-\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$

Answer

$\frac{1}{x^2}\int\ln\left(x\right)^2dx+\int\int\int\ln\left(x\right)^2dxdxdxcosh\left(x\right)+\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxcosh\left(x\right)+\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxcosh\left(x\right)+\int cosh\left(x\right)\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxdx-\int\int\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\int\int\ln\left(x\right)^2dxdxdxdxdxdxcosh\left(x\right)-\int\int\int\int\ln\left(x\right)^2dxdxdxdxcosh\left(x\right)-\int\int\ln\left(x\right)^2dxdxcosh\left(x\right)$

Problem Analysis

Main topic:

Integration by parts

Time to solve it:

1.5 seconds

Views:

104