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Step-by-step Solution

Solve the trigonometric integral $\int x^4\ln\left(5x\right)dx$

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Answer

$x^{5}\left(-\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}\ln\left(5x\right)\right)+\frac{33}{\sqrt[3]{3}}x^{5}+C_0$

Step-by-step explanation

Problem to solve:

$\int_{ }^{ }x^4\ln\left(5x\right)dx$
1

Solve the integral $\int x^4\ln\left(5x\right)dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=5x \\ du=5dx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{5}=dx$

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Answer

$x^{5}\left(-\frac{3}{\sqrt{2}}+\frac{3}{\sqrt{2}}\ln\left(5x\right)\right)+\frac{33}{\sqrt[3]{3}}x^{5}+C_0$