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

Problem 4.257 · 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. \[ = \int \frac{\sin{\left(e^{2 x - 1} \right)} \frac{d}{d x} e^{2 x - 1}}{2}\, dx \]
   rewriteRewrite the integrand to highlight the derivative of the inner function.✓ Proved
3. \[ = \int e^{2 x - 1} \sin{\left(e^{2 x - 1} \right)}\, dx \]
   simplify simplifySimplify the expression inside the integral. Simplify the constant factor.✓ Proved
4. \[ = - \frac{\cos{\left(e^{2 x - 1} \right)}}{2} \]
   antiderivativeIntegrate using the substitution u = exp(2*x - 1), where du = 2*exp(2*x - 1)dx.✓ Proved