3.2 Trigonometric integrals  (Page 2/6)

Problem-solving strategy: integrating ${{\displaystyle \int }}^{}{\text{tan}}^{k}x{\text{sec}}^{j}xdx$

To integrate ${{\displaystyle \int }}^{}{\text{tan}}^{k}x{\text{sec}}^{j}xdx,$ use the following strategies:

1. If $j$ is even and $j\ge 2,$ rewrite ${\text{sec}}^{j}x={\text{sec}}^{j-2}x{\text{sec}}^{2}x$ and use ${\text{sec}}^{2}x={\text{tan}}^{2}x+1$ to rewrite ${\text{sc}}^{j-2}x$ in terms of $\text{tan}x.$ Let $u=\text{tan}x$ and $du={\text{sec}}^{2}x.$
2. If $k$ is odd and $j\ge 1,$ rewrite ${\text{tan}}^{k}x{\text{sec}}^{j}x={\text{tan}}^{k-1}x{\text{sec}}^{j-1}x\text{sec}x\text{tan}x$ and use ${\text{tan}}^{2}x={\text{sec}}^{2}x-1$ to rewrite ${\text{tan}}^{k-1}x$ in terms of $\text{sec}x.$ Let $u=\text{sec}x$ and $du=\text{sec}x\text{tan}xdx.$ ( Note : If $j$ is even and $k$ is odd, then either strategy 1 or strategy 2 may be used.)
3. If $k$ is odd where $k\ge 3$ and $j=0,$ rewrite ${\text{tan}}^{k}x={\text{tan}}^{k-2}x{\text{tan}}^{2}x={\text{tan}}^{k-2}x({\text{sec}}^{2}x-1)={\text{tan}}^{k-2}x{\text{sec}}^{2}x-{\text{tan}}^{k-2}x.$ It may be necessary to repeat this process on the ${\text{tan}}^{k-2}x$ term.
4. If $k$ is even and $j$ is odd, then use ${\text{tan}}^{2}x={\text{sec}}^{2}x-1$ to express ${\text{tan}}^{k}x$ in terms of $\text{sec}x.$ Use integration by parts to integrate odd powers of $\text{sec}x.$