Exercise 11

This problem centers on finding antiderivatives for the basic trigonometric functions other than $\sin(x)$ and $\cos(x)$.

1. Consider the indefinite integral $\int \tan(x) \, dx$. By rewriting the integrand as $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and identifying an appropriate function-derivative pair, make a $u$-substitution and hence evaluate $\int \tan(x) \, dx$.

2. In a similar way, evaluate $\int \cot(x) \, dx$.

3. Consider the indefinite integral $$\begin{gathered} \int \frac{\sec^2(x) + \sec(x) \tan(x)}{\sec(x) + \tan(x)} \, dx \end{gathered}$$. Evaluate this integral using the substitution $u = \sec(x) + \tan(x)$.

4. Simplify the integrand in (c) by factoring the numerator. What is a far simpler way to write the integrand?

5. Combine your work in (c) and (d) to determine $\int \sec(x) \, dx$.

6. Using (c)-(e) as a guide, evaluate $\int \csc(x) \, dx$.