5. \( \int_{-1}^{0} 2 x \cdot e^{-x} d x \) есептеңіз.

Calculate the integral \( \int_{-1}^{0} 2x*e^(-x) \, dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int_{-1}^{0} 2xe^{-x} dx\) - step1: Evaluate the integral: \(\int 2xe^{-x} dx\) - step2: Use properties of integrals: \(2\times \int xe^{-x} dx\) - step3: Prepare for integration by parts: \(\begin{align}&u=x\\&dv=e^{-x}dx\end{align}\) - step4: Calculate the derivative: \(\begin{align}&du=dx\\&dv=e^{-x}dx\end{align}\) - step5: Evaluate the integral: \(\begin{align}&du=dx\\&v=-e^{-x}\end{align}\) - step6: Substitute the values into formula: \(2\left(x\left(-e^{-x}\right)-\int 1\times \left(-e^{-x}\right) dx\right)\) - step7: Calculate: \(2\left(-xe^{-x}-\int -e^{-x} dx\right)\) - step8: Calculate: \(-2xe^{-x}-2\times \int -e^{-x} dx\) - step9: Evaluate the integral: \(-2xe^{-x}-2e^{-x}\) - step10: Return the limits: \(\left(-2xe^{-x}-2e^{-x}\right)\bigg |_{-1}^{0}\) - step11: Calculate the value: \(-2\) Результат интеграла \( \int_{-1}^{0} 2x \cdot e^{-x} dx \) равен -2.