Question

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

$int e^{(-sin x)} (-cos x) , dx = square$

Explanation:

Step1: Choose substitution

Let \( u = -\sin x \). Then, find the derivative of \( u \) with respect to \( x \).
The derivative of \( -\sin x \) with respect to \( x \) is \( du = -\cos x \, dx \).

Step2: Substitute into integral

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

Answer:

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