Primitive of Square of Tangent Function/Proof 1
Jump to navigation
Jump to search
Theorem
- $\ds \int \tan^2 x \rd x = \tan x - x + C$
Proof
| \(\ds \int \tan^2 x \rd x\) | \(=\) | \(\ds \int \paren {\sec^2 x - 1} \rd x\) | Difference of Squares of Secant and Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \sec^2 x \rd x + \int \paren {-1} \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tan x + C + \int \paren {-1} \rd x\) | Primitive of Square of Secant Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tan x - x + C\) | Primitive of Constant |
$\blacksquare$