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

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Solution

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Final Answer

$\frac{19}{175}x^{4}\ln\left(x\right)-\frac{6}{221}x^{4}+C_0$
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Step-by-step Solution

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

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

Learn how to solve differential equations problems step by step online.

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

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Learn how to solve differential equations 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)}. Simplify the expression inside the integral. The integral of a function times a constant (\frac{76}{175}) is equal to the constant times the integral of the function. 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

$\frac{19}{175}x^{4}\ln\left(x\right)-\frac{6}{221}x^{4}+C_0$

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 integral of x^3log(10,x)dx using partial fractionsSolve integral of x^3log(10,x)dx using basic integralsSolve integral of x^3log(10,x)dx using u-substitutionSolve integral of x^3log(10,x)dx using integration by partsSolve integral of x^3log(10,x)dx using tabular integrationSolve integral of x^3log(10,x)dx using trigonometric substitution

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Plotting: $\frac{19}{175}x^{4}\ln\left(x\right)-\frac{6}{221}x^{4}+C_0$

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x
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.
(◻)
+
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×
◻/◻
/
÷
◻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

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Main Topic: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.

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