Primitive of Cube of Tangent of a x/Proof 1
Jump to navigation
Jump to search
Theorem
- $\ds \int \tan^3 a x \rd x = \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C$
Proof
| \(\ds \int \tan^3 a x \rd x\) | \(=\) | \(\ds \int \tan a x \tan^2 a x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \tan a x \paren {\sec^2 a x - 1} \rd x\) | Difference of Squares of Secant and Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \tan a x \sec^2 a x \rd x - \int \tan a x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan^2 a x} {2 a} - \int \tan a x \rd x + C\) | Primitive of $\tan^n a x \sec^2 a x$: $n = 1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan^2 a x} {2 a} - \paren {\frac {-\ln \size {\cos a x} } a} + C\) | Primitive of $\tan a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\tan^2 a x} {2 a} + \frac 1 a \ln \size {\cos a x} + C\) | simplifying |
$\blacksquare$