Primitive of Power of Tangent of a x
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Theorem
- $\ds \int \tan^n a x \rd x = \frac {\tan^{n - 1} a x} {\paren {n - 1} a} - \int \tan^{n - 2} a x \rd x + C$
Proof
| \(\ds \int \tan^n a x \rd x\) | \(=\) | \(\ds \int \tan^{n - 2} a x \tan^2 a x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \tan^{n - 2} a x \paren {\sec^2 a x - 1} \rd x\) | Difference of Squares of Secant and Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \tan^{n - 2} a x \sec^2 a x \rd x - \int \tan^{n - 2} \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan^{n - 1} a x} {\paren {n - 1} a} - \int \tan^{n - 2} \rd x + C\) | Primitive of $\tan^n a x \sec^2 a x$ |
$\blacksquare$
Also see
- Primitive of $\sin^n a x$
- Primitive of $\cos^n a x$
- Primitive of $\cot^n a x$
- Primitive of $\sec^n a x$
- Primitive of $\csc^n a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tan a x$: $14.439$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae