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Solve the integral of logarithmic functions $\int x^2\ln\left(x\right)dx$

Step-by-step Solution

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Solution

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Final Answer

$\frac{x^{3}\ln\left(x\right)}{3}+\frac{-x^{3}}{9}+C_0$
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Step-by-step Solution

Problem to solve:

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

Specify the solving method

1

We can solve the integral $\int x^2\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify $u$ and calculate $du$

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

Learn how to solve simplification of algebraic expressions problems step by step online.

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

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Learn how to solve simplification of algebraic expressions problems step by step online. Solve the integral of logarithmic functions int(x^2ln(x))dx. We can solve the integral \int x^2\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. First, identify u and calculate du. Now, identify dv and calculate v. Solve the integral.

Final Answer

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

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

Solve int(x^2ln(x))dx using basic integralsSolve int(x^2ln(x))dx using u-substitutionSolve int(x^2ln(x))dx using integration by partsSolve int(x^2ln(x))dx using tabular integration
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Go!
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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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

How to improve your answer:

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

Main topic:

Simplification of algebraic expressions

Time to solve it:

~ 0.13 s

Related topics:

Simplification of algebraic expressionsAlgebra

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