Question

use substitution to evaluate the indefinite integral\\(\int e^{(-\cos x)} \sin x \\, dx\\)\\(\int e^{(-\cos x)} \sin x \\, dx = \square\\)

Explanation:

Step1: Choose substitution

Let \( u = -\cos x \). Then, find the derivative of \( u \) with respect to \( x \). Using the derivative rule for \( \cos x \), we have \( \frac{du}{dx}=\sin x \). Rearranging, we get \( du = \sin x \, dx \).

Step2: Substitute into integral

The original integral is \( \int e^{-\cos x} \sin x \, dx \). Substituting \( u = -\cos x \) and \( du = \sin x \, dx \), the integral becomes \( \int e^{u} \, du \).

Step3: Integrate \( e^u \)

The integral of \( e^u \) with respect to \( u \) is \( e^u + C \), where \( C \) is the constant of integration.

Step4: Substitute back \( u \)

Substitute \( u = -\cos x \) back into the result. So we get \( e^{-\cos x}+C \).

Answer:

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