Primitive of Tangent Function/Cosine Form/Proof
Theorem
- $\ds \int \tan x \rd x = -\ln \size {\cos x} + C$
where $\cos x \ne 0$.
Proof
| \(\ds \int \tan x \rd x\) | \(=\) | \(\ds \int \frac {\sin x} {\cos x} \rd x\) | Definition of Real Tangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\int \frac {-\sin x} {\cos x} \rd x\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\int \frac {\paren {\cos x}'} {\cos x} \rd x\) | Derivative of Cosine Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\ln \size {\cos x} + C\) | Primitive of Function under its Derivative |
$\blacksquare$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Example