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# Solve the integral of logarithmic functions $\int x^3\log \left(x\right)^2dx$

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##  Final answer to the problem

$\frac{8x^{4}\ln\left|x\right|^2+x^{4}-4x^{4}\ln\left|x\right|}{32\ln\left|10\right|^2}+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Integrate by partial fractions
• Integrate by substitution
• Integrate by parts
• Integrate using tabular integration
• Integrate by trigonometric substitution
• Weierstrass Substitution
• Integrate using trigonometric identities
• Integrate using basic integrals
• Product of Binomials with Common Term
Can't find a method? Tell us so we can add it.
1

Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\int x^3\left(\frac{\ln\left(x\right)}{\ln\left(10\right)}\right)^2dx$

Learn how to solve integrals of exponential functions problems step by step online.

$\int x^3\left(\frac{\ln\left(x\right)}{\ln\left(10\right)}\right)^2dx$

Learn how to solve integrals of exponential functions problems step by step online. Solve the integral of logarithmic functions int(x^3log(x)^2)dx. Change the logarithm to base e applying the change of base formula for logarithms: \log_b(a)=\frac{\log_x(a)}{\log_x(b)}. Simplify the expression. Take the constant \frac{1}{\ln\left|10\right|^2} out of the integral. We can solve the integral \int\ln\left(x\right)^2x^3dx by applying integration by parts method to calculate the integral of the product of two functions, using the following formula.

##  Final answer to the problem

$\frac{8x^{4}\ln\left|x\right|^2+x^{4}-4x^{4}\ln\left|x\right|}{32\ln\left|10\right|^2}+C_0$

##  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

SnapXam A2

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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: Integrals of Exponential Functions

Those are integrals that involve exponential functions. Recall that an exponential function is a function of the form f(x)=a^x.