To solve the integral \( \int \sec(x) \sin(x) , dx \), we can use a substitution method.
Recall that \( \sec(x) \) can be written as \( \frac{1}{\cos(x)} \). Therefore, we can rewrite the integral as:
\[ \int \sec(x) \sin(x) , dx = \int \frac{\sin(x)}{\cos(x)} , dx = \int \tan(x) , dx \]
The integral of \( \tan(x) \) is:
\[ \int \tan(x) , dx = -\ln |\cos(x)| + C \]
where \( C \) is the constant of integration.
Thus, the final answer is:
\[ \int \sec(x) \sin(x) , dx = -\ln |\cos(x)| + C \]
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