Mathis Nunez
09/20/2023 · Elementary School
4. \( \int(x) \arctan (x) d x \)
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Step-by-step Solution
Calculate the integral \( \int(x) \arctan (x) dx \).
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
\(\int x\arctan\left(x\right) dx\)
- step1: Prepare for integration by parts:
\(\begin{align}&u=\arctan\left(x\right)\\&dv=xdx\end{align}\)
- step2: Calculate the derivative:
\(\begin{align}&du=\frac{1}{1+x^{2}}dx\\&dv=xdx\end{align}\)
- step3: Evaluate the integral:
\(\begin{align}&du=\frac{1}{1+x^{2}}dx\\&v=\frac{1}{2}x^{2}\end{align}\)
- step4: Substitute the values into formula:
\(\arctan\left(x\right)\times \frac{1}{2}x^{2}-\int \frac{1}{1+x^{2}}\times \frac{1}{2}x^{2} dx\)
- step5: Calculate:
\(\frac{1}{2}x^{2}\arctan\left(x\right)-\int \frac{x^{2}}{2+2x^{2}} dx\)
- step6: Evaluate the integral:
\(\frac{1}{2}x^{2}\arctan\left(x\right)-\frac{1}{2}x+\frac{1}{2}\arctan\left(x\right)\)
- step7: Add the constant of integral C:
\(\frac{1}{2}x^{2}\arctan\left(x\right)-\frac{1}{2}x+\frac{1}{2}\arctan\left(x\right) + C, C \in \mathbb{R}\)
La integral de \( \int(x) \arctan (x) dx \) es igual a \( \frac{1}{2}x^{2}\arctan(x)-\frac{1}{2}x+\frac{1}{2}\arctan(x) + C \), donde \( C \) es una constante real.
Quick Answer
La integral de \( \int(x) \arctan (x) dx \) es \( \frac{1}{2}x^{2}\arctan(x)-\frac{1}{2}x+\frac{1}{2}\arctan(x) + C \).
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