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

Problem 4.172 · medium

1. \[ \int \left(2 x + 1\right)^{2} e^{\left(2 x + 1\right)^{3}}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(2 x + 1\right)^{2} e^{2 x^{3} + 3 x^{2} + 3 x + 1}\, dx \]
   algebraExpand the exponent.Not checked
3. \[ = \frac{\int 3 \left(2 x + 1\right)^{2} e^{\left(2 x + 1\right)^{3}}\, dx}{3} \]
   rewriteThis is not a good way to proceed. Let's use the chain rule in reverse.Not checked
4. \[ = \int \left(2 x + 1\right)^{2} e^{\left(2 x + 1\right)^{3}}\, dx \]
   rewrite algebraNo change needed, let's identify the derivative of the exponent. Express the integrand in a form suitable for substitution.✓ Proved
5. \[ = \int \frac{3 \left(2 x^{2} + \frac{4 x}{3} + \frac{2}{3}\right) e^{\left(2 x + 1\right)^{3}}}{2 \left(2 x + 1\right)^{2} \left(3 x^{2} + 2 x + 1\right)}\, dx \]
   algebraActually, let's just differentiate the exponent.Not checked
6. \[ = \int \left(2 x + 1\right)^{2} e^{\left(2 x + 1\right)^{3}}\, dx \]
   rewrite algebra algebraThe derivative of (2*x + 1)**3 is 3 * (2*x + 1)**2 * 2 = 6 * (2*x + 1)**2. Rewrite the integrand to include the derivative of the exponent. Group the terms.Not checked
7. \[ = \frac{e^{\left(2 x + 1\right)^{3}}}{6} \]
   antiderivativeThe integral is the reciprocal of the constant factor times the exponential part.✓ Proved