Question
integral of (secx)(sinx)
Answers
GPT-4o mini
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:
\[
\int \sec(x) \sin(x) \, dx = -\ln |\cos(x)| + C
\]
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:
\[
\int \sec(x) \sin(x) \, dx = -\ln |\cos(x)| + C
\]