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 variable

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

Step2: Substitute into the integral

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

Step3: Integrate with respect to \( u \)

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

Step4: Substitute back \( u = -\cos x \)

Replacing \( u \) with \( -\cos x \), we get \( e^{-\cos x} + C \).

Answer:

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