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

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Final answer to the problem

$\frac{4x^{4}\ln\left|x\right|-x^{4}}{16\ln\left|10\right|}+C_0$
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Step-by-step Solution

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  • Integrate by partial fractions
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  • Product of Binomials with Common Term
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Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

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

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

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Learn how to solve factorization problems step by step online. Solve the integral of logarithmic functions int(x^3log(x))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)}. Multiplying the fraction by x^3. Take the constant \frac{1}{\ln\left|10\right|} out of the integral. We can solve the integral \int x^3\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.

Final answer to the problem

$\frac{4x^{4}\ln\left|x\right|-x^{4}}{16\ln\left|10\right|}+C_0$

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Function Plot

Plotting: $\frac{4x^{4}\ln\left(x\right)-x^{4}}{16\ln\left(10\right)}+C_0$

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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

How to improve your answer:

Main Topic: Factorization

In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original.

Used Formulas

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