Question

evaluate the indefinite integral. \\(\int 3e^{3x} \sin(e^{3x}) \\, dx\\) \\(\boxed{\quad} + c\\)

Explanation:

Step1: Choose substitution variable

Let $u = e^{3x}$.

Step2: Compute derivative of u

$du = 3e^{3x}dx$.

Step3: Substitute into integral

Integral becomes $\int \sin(u) du$.

Step4: Integrate sin(u)

$\int \sin(u) du = -\cos(u) + C$.

Step5: Substitute back u

Replace u with $e^{3x}$: $-\cos(e^{3x}) + C$.

Answer:

$-\cos(e^{3x})$