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 \( x \).
The derivative of \( \cos x \) 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 \( du = -\sin x \, dx \). Substituting these into the integral, we get:
\( \int e^{u} \, du \)

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 \( u \)

Now, substitute back \( u = \cos x \) into the result. So we have \( e^{\cos x} + C \).

Answer:

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