Integral of \( \displaystyle \left(3 x - 1\right) e^{3 x - 1} \)

Problem 4.169 · medium

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