Primitive of Tangent Function/Secant Form/Proof
Theorem
- $\ds \int \tan x \rd x = \ln \size {\sec x} + C$
where $\sec x$ is defined.
Proof
| \(\ds \int \tan x \rd x\) | \(=\) | \(\ds -\ln \size {\cos x} + C\) | Primitive of $\tan x$: Cosine Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\frac 1 {\cos x} } + C\) | Logarithm of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\sec x} + C\) | Secant is Reciprocal of Cosine |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration