Burton Sullivan
08/14/2024 · High School
Find the value of \( \int \log _{e} x \cdot d x \)
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Step-by-step Solution
Calculate the integral \( \int \log _{e} x \cdot d x \).
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
\(\int \ln{\left(x\right)} dx\)
- step1: Prepare for integration by parts:
\(\begin{align}&u=\ln{\left(x\right)}\\&dv=dx\end{align}\)
- step2: Calculate the derivative:
\(\begin{align}&du=\frac{1}{x}dx\\&dv=dx\end{align}\)
- step3: Evaluate the integral:
\(\begin{align}&du=\frac{1}{x}dx\\&v=x\end{align}\)
- step4: Substitute the values into formula:
\(\ln{\left(x\right)}\times x-\int \frac{1}{x}\times x dx\)
- step5: Calculate:
\(x\ln{\left(x\right)}-\int 1 dx\)
- step6: Evaluate the integral:
\(x\ln{\left(x\right)}-x\)
- step7: Add the constant of integral C:
\(x\ln{\left(x\right)}-x + C, C \in \mathbb{R}\)
The value of \( \int \log _{e} x \cdot d x \) is \( x\ln{(x)}-x + C \), where \( C \) is an arbitrary constant.
Quick Answer
\( x\ln{(x)}-x + C \)
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