Integral of \( \displaystyle \left(3 x + 1\right)^{2} e^{3 x + 1} \)

Problem 4.729 · medium

1. \[ \int \left(3 x + 1\right)^{2} e^{3 x + 1}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(9 x^{2} + 6 x + 1\right) e^{3 x + 1}\, dx \]
   algebraExpand the squared binomial.✓ Proved
3. \[ = 6 \int x e^{3 x + 1}\, dx + 9 \int x^{2} e^{3 x + 1}\, dx + \int e^{3 x + 1}\, dx \]
   linearityDistribute the exponential term and split the integral into three parts.✓ Proved
4. \[ = \frac{e^{3 x + 1}}{3} + 6 \int x e^{3 x + 1}\, dx + 9 \int x^{2} e^{3 x + 1}\, dx \]
   antiderivativeEvaluate the integral of the constant term.✓ Proved
5. \[ = 3 x^{2} e^{3 x + 1} + \frac{e^{3 x + 1}}{3} - 9 \int \frac{2 x e^{3 x + 1}}{3}\, dx + 6 \int x e^{3 x + 1}\, dx \]
   partsApply integration by parts to the first integral term.✓ Proved
6. \[ = 3 x^{2} e^{3 x + 1} + \frac{e^{3 x + 1}}{3} \]
   simplify algebra simplifySimplify the expression inside the parentheses. Distribute the 9 into the parentheses. Cancel the remaining integral terms.✓ Proved
7. \[ = \left(3 x^{2} + \frac{1}{3}\right) e^{3 x + 1} \]
   algebraFactor out the exponential term.✓ Proved