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 \( du \) by differentiating \( u \) with respect to \( x \). The derivative of \( -\cos x \) with respect to \( x \) is \( \sin x \), so \( du = \sin x \, dx \).

Step2: Substitute into integral

The integral \( \int e^{-\cos x} \sin x \, dx \) can be rewritten using the substitution \( u = -\cos x \) and \( du = \sin x \, dx \) as \( \int e^{u} \, du \).

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

Answer:

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