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

Problem 4.152 · medium

1. \[ \int \left(2 x - 1\right)^{2} e^{2 x - 1}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(4 x^{2} - 4 x + 1\right) e^{2 x - 1}\, dx \]
   algebraExpand the squared binomial.✓ Proved
3. \[ = - \int 4 x e^{2 x - 1}\, dx + \int 4 x^{2} e^{2 x - 1}\, dx + \int e^{2 x - 1}\, dx \]
   linearityDistribute the exponential term and split the integral.✓ Proved
4. \[ = \frac{e^{2 x - 1}}{2} - \int 4 x e^{2 x - 1}\, dx + \int 4 x^{2} e^{2 x - 1}\, dx \]
   antiderivativeIntegrate the constant term.✓ Proved
5. \[ = - 2 x e^{2 x - 1} + \frac{e^{2 x - 1}}{2} + \int 4 x^{2} e^{2 x - 1}\, dx + \int 2 e^{2 x - 1}\, dx \]
   parts algebraApply integration by parts to the second term. Simplify the result of integration by parts.✓ Proved
6. \[ = - 2 x e^{2 x - 1} + \frac{3 e^{2 x - 1}}{2} + \int 4 x^{2} e^{2 x - 1}\, dx \]
   antiderivative simplifyIntegrate the remaining integral. Combine the exponential terms.✓ Proved
7. \[ = 2 x^{2} e^{2 x - 1} - 2 x e^{2 x - 1} + \frac{3 e^{2 x - 1}}{2} - \int 4 x e^{2 x - 1}\, dx \]
   partsApply integration by parts to the first term.✓ Proved
8. \[ = 2 x^{2} e^{2 x - 1} - 4 x e^{2 x - 1} + \frac{3 e^{2 x - 1}}{2} + \int 2 e^{2 x - 1}\, dx \]
   partsApply integration by parts to the integral of x*exp(2x-1).✓ Proved
9. \[ = 2 x^{2} e^{2 x - 1} - 4 x e^{2 x - 1} + \frac{5 e^{2 x - 1}}{2} \]
   antiderivative simplifyIntegrate the remaining exponential term. Combine all terms into the final antiderivative.✓ Proved