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

Problem 4.798 · medium

1. \[ \int e^{2 x + 1} \sin{\left(e^{2 x + 1} \right)}\, dx \]
   integral algebraStart with the integral of the given function. Multiply and divide by 2 to prepare for substitution.✓ Proved
2. \[ = \int \frac{\sin{\left(e^{2 x + 1} \right)} \frac{d}{d x} e^{2 x + 1}}{2}\, dx \]
   rewriteExpress the integrand in terms of the derivative of the inner function.✓ Proved
3. \[ = \int e^{2 x + 1} \sin{\left(e^{2 x + 1} \right)}\, dx \]
   simplify simplifyCalculate the derivative of the inner function. Simplify the expression inside the integral.✓ Proved
4. \[ = - \frac{\cos{\left(e^{2 x + 1} \right)}}{2} \]
   antiderivativeIntegrate using the substitution u = exp(2*x + 1).✓ Proved