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# Solve the integral of logarithmic functions $\int\sqrt{x}\ln\left(x\right)dx$

## Step-by-step Solution

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###  Videos

$\frac{2}{3}\sqrt{x^{3}}\ln\left(x\right)-\frac{4}{9}\sqrt{x^{3}}+C_0$
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##  Step-by-step Solution 

Problem to solve:

$\int\sqrt{x}\ln\left(x\right)dx$

Specify the solving method

1

We can solve the integral $\int\sqrt{x}\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

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

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{x}$
2

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\ln\left(x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\sqrt{x}dx}\\ \displaystyle{\int dv=\int \sqrt{x}dx}\end{matrix}$
4

Solve the integral

$v=\int\sqrt{x}dx$
5

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $\frac{1}{2}$

$\frac{1}{\frac{3}{2}}\sqrt{x^{3}}$

The integral of a function times a constant ($\frac{2}{3}$) is equal to the constant times the integral of the function

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

Multiplying the fraction by $\sqrt{x^{3}}$

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

Simplify the fraction $\frac{\sqrt{x^{3}}}{x}$ by $x$

$\frac{2}{3}\sqrt{x^{3}}\ln\left(x\right)-\frac{2}{3}\int\sqrt{x}dx$
6

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

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

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $\frac{1}{2}$

$-\frac{2}{3}\cdot \left(\frac{1}{\frac{3}{2}}\right)\sqrt{x^{3}}$

Simplify the expression inside the integral

$-\frac{4}{9}\sqrt{x^{3}}$
7

The integral $-\frac{2}{3}\int\sqrt{x}dx$ results in: $-\frac{4}{9}\sqrt{x^{3}}$

$-\frac{4}{9}\sqrt{x^{3}}$
8

Gather the results of all integrals

$\frac{2}{3}\sqrt{x^{3}}\ln\left(x\right)-\frac{4}{9}\sqrt{x^{3}}$
9

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{2}{3}\sqrt{x^{3}}\ln\left(x\right)-\frac{4}{9}\sqrt{x^{3}}+C_0$

$\frac{2}{3}\sqrt{x^{3}}\ln\left(x\right)-\frac{4}{9}\sqrt{x^{3}}+C_0$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

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SnapXam A2

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1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Main topic:

Differential Calculus

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