Sure, let's solve the integral [tex]\(\int \sec^3(x) \csc(x) \, dx\)[/tex] step-by-step.
1. Identify the expression to integrate:
[tex]\[
\int \sec^3(x) \csc(x) \, dx
\][/tex]
2. Simplify the integrand:
To approach this integral, we start by breaking down the integrand. The integrand is [tex]\(\sec^3(x) \csc(x)\)[/tex]. Recall that [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex] and [tex]\(\csc(x) = \frac{1}{\sin(x)}\)[/tex]. Thus:
[tex]\[
\sec^3(x) \csc(x) = \left(\frac{1}{\cos(x)}\right)^3 \cdot \frac{1}{\sin(x)} = \frac{1}{\cos^3(x) \sin(x)}
\][/tex]
3. Substitute and integrate:
To integrate this expression, we perform a substitution method. However, since this could become quite complex, it's often more practical to use known integrals or computational tools to find the antiderivative.
The integration of this specific function yields the following result:
[tex]\[
\int \sec^3(x) \csc(x) \, dx = \frac{\log(\cos^2(x) - 1)}{2} - \log(\cos(x)) + \frac{1}{2\cos^2(x)}
\][/tex]
4. Include the constant of integration:
Finally, don't forget to add the constant of integration [tex]\(C\)[/tex], since we are integrating with respect to [tex]\(x\)[/tex]:
[tex]\[
\frac{\log(\cos^2(x) - 1)}{2} - \log(\cos(x)) + \frac{1}{2\cos^2(x)} + C
\][/tex]
So, the detailed step-by-step solution to the integral [tex]\(\int \sec^3(x) \csc(x) \, dx\)[/tex] is:
[tex]\[
\int \sec^3(x) \csc(x) \, dx = \frac{\log(\cos^2(x) - 1)}{2} - \log(\cos(x)) + \frac{1}{2\cos^2(x)} + C
\][/tex]
