Primitive of Square of Tangent Function
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Theorem
- $\ds \int \tan^2 x \rd x = \tan x - x + C$
where $C$ is an arbitrary constant.
Proof 1
| \(\ds \int \tan^2 x \rd x\) | \(=\) | \(\ds \int \paren {\sec^2 x - 1} \rd x\) | 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 |
$\blacksquare$
Proof 2
| \(\ds I_n\) | \(=\) | \(\ds \int \tan^n x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan^{n - 1} x} {n - 1} - I_{n - 2}\) | Reduction Formula for Integral of Power of Tangent | |||||||||||
| \(\ds I_0\) | \(=\) | \(\ds \int \paren {\tan x}^0 \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \d x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x + C\) | Primitive of Constant | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds I_2\) | \(=\) | \(\ds \tan x - x + C'\) | setting $n = 2$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.19$
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals