\(\int e^x\sin\left(e^x\right)dx\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-\cos\left(e^x\right)+C\)
The factor \(e^x\) is the derivative of the inner function \(e^x\), so the substitution \(t = e^x\) works.
\( t = e^x,\quad dt = e^x\,dx \)
\( \int e^x\sin\left(e^x\right)dx = \int\sin t\,dt \)
\( = -\cos t+C \)
\( = -\cos\left(e^x\right)+C \)