Integral of \( \displaystyle \left(2 x - 1\right) e^{1 - 2 x} \)

Problem 4.143 · medium

1. \[ \int \left(2 x - 1\right) e^{1 - 2 x}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(2 x e^{1 - 2 x} - e^{1 - 2 x}\right)\, dx \]
   linearityDistribute the exponential term.✓ Proved
3. \[ = 2 \int x e^{1 - 2 x}\, dx - \int e^{1 - 2 x}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = \frac{e^{1 - 2 x}}{2} + 2 \int x e^{1 - 2 x}\, dx \]
   antiderivativeEvaluate the second integral.✓ Proved
5. \[ = - x e^{1 - 2 x} + \frac{e^{1 - 2 x}}{2} - 2 \int \left(- \frac{e^{1 - 2 x}}{2}\right)\, dx \]
   partsApply integration by parts to the first integral.✓ Proved
6. \[ = - x e^{1 - 2 x} + \frac{e^{1 - 2 x}}{2} + \int e^{1 - 2 x}\, dx \]
   algebra algebraSimplify the expression inside the parentheses. Distribute the 2.✓ Proved
7. \[ = - x e^{1 - 2 x} \]
   antiderivative simplifyEvaluate the remaining integral. Combine the terms.✓ Proved