Integral of \( \displaystyle 9 x^{2} e^{3 x} \)

Problem 4.560 · medium

1. \[ \int 9 x^{2} e^{3 x}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = 9 \int x^{2} e^{3 x}\, dx \]
   linearityFactor out the constant 9.✓ Proved
3. \[ = 3 x^{2} e^{3 x} - 3 \int 2 x e^{3 x}\, dx \]
   parts algebraApply integration by parts with u = x**2 and dv = exp(3*x)dx. Distribute the 9 and simplify the fraction.✓ Proved
4. \[ = 3 x^{2} e^{3 x} - 6 \int x e^{3 x}\, dx \]
   algebraFactor out the 2 from the integral.✓ Proved
5. \[ = 3 x^{2} e^{3 x} - 2 x e^{3 x} + 2 \int e^{3 x}\, dx \]
   parts algebraApply integration by parts to the remaining integral with u = x and dv = exp(3*x)dx. Distribute the -6 and simplify the terms.✓ Proved
6. \[ = 3 x^{2} e^{3 x} - 2 x e^{3 x} + \frac{2 e^{3 x}}{3} \]
   antiderivativeEvaluate the final integral.✓ Proved
7. \[ = \left(3 x^{2} - 2 x + \frac{2}{3}\right) e^{3 x} \]
   simplifyFactor out the common exponential term.✓ Proved