The Notes: `int tan^2(x) sec(x) dx` Evaluate the integral

$\displaystyle\int{{\tan}}^{{2}}{\left({x}\right)}{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}$

use the identity: $\displaystyle{{\tan}}^{{2}}{\left({x}\right)}={{\sec}}^{{2}}{\left({x}\right)}-{1}$

$\displaystyle\int{{\tan}}^{{2}}{\left({x}\right)}{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}=\int{\left({{\sec}}^{{2}}{\left({x}\right)}-{1}\right)}{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}$

$\displaystyle=\int{\left({{\sec}}^{{3}}{\left({x}\right)}-{\sec{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$

Now apply the Integral Reduction: $\displaystyle\int{{\sec}}^{{n}}{\left({x}\right)}{\left.{d}{x}\right.}=\frac{{{{\sec}}^{{{n}-{1}}}{\left({x}\right)}{\sin{{\left({x}\right)}}}}}{{{n}-{1}}}+\frac{{{n}-{2}}}{{{n}-{1}}}\int{{\sec}}^{{{n}-{2}}}{\left({x}\right)}{\left.{d}{x}\right.}$

$\displaystyle\int{{\sec}}^{{3}}{\left({x}\right)}{\left.{d}{x}\right.}=\frac{{{{\sec}}^{{2}}{\left({x}\right)}{\sin{{\left({x}\right)}}}}}{{2}}+\frac{{1}}{{2}}\int{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}$

Use the common integral: $\displaystyle\int{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}={\ln{{\left({\tan{{\left({x}\right)}}}+{\sec{{\left({x}\right)}}}\right)}}}$

$\displaystyle\therefore\int{{\tan}}^{{2}}{\left({x}\right)}{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}=\frac{{{{\sec}}^{{2}}{\left({x}\right)}{\sin{{\left({x}\right)}}}}}{{2}}+\frac{{1}}{{2}}\int{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}-\int{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}$

$\displaystyle=\frac{{{{\sec}}^{{2}}{\left({x}\right)}{\sin{{\left({x}\right)}}}}}{{2}}-\frac{{1}}{{2}}\int{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}$

$\displaystyle=\frac{{{{\sec}}^{{2}}{\left({x}\right)}{\sin{{\left({x}\right)}}}}}{{2}}-\frac{{1}}{{2}}{\ln{{\left({\tan{{\left({x}\right)}}}+{\sec{{\left({x}\right)}}}\right)}}}$

add a constant C to the solution,

$\displaystyle=\frac{{{{\sec}}^{{2}}{\left({x}\right)}{\sin{{\left({x}\right)}}}}}{{2}}-\frac{{1}}{{2}}{\ln{{\left({\tan{{\left({x}\right)}}}+{\sec{{\left({x}\right)}}}\right)}}}+{C}$

The Notes

Thursday, April 21, 2016

$\displaystyle\int{{\tan}}^{{2}}{\left({x}\right)}{\sec{{\left({x}\right)}}}{\left.{d}{x}\right.}$ Evaluate the integral

No comments:

Post a Comment

What are hearing tests?