Question

evaluate the indefinite integral.\\(int 4e^{4x}sin(e^{4x}) dx\\)\\(\boxed{}\\) + c

Explanation:

Step1: Choose substitution variable

Let $u = e^{4x}$

Step2: Compute derivative of u

$\frac{du}{dx} = 4e^{4x} \implies du = 4e^{4x}dx$

Step3: Rewrite integral in terms of u

$\int \sin(u) du$

Step4: Integrate $\sin(u)$

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

Step5: Substitute back $u = e^{4x}$

$-\cos(e^{4x}) + C$

Answer:

$-\cos(e^{4x})$