Primitive of Cube of Secant Function
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Theorem
- $\ds \int \sec^3 x \rd x = \frac 1 2 \paren {\sec x \tan x + \ln \size {\sec x + \tan x} } + C$
Proof 1
From Primitive of $\sec^3 a x$:
- $\ds \int \sec^3 a x \rd x = \frac 1 {2 a} \paren {\sec a x \tan a x + \ln \size {\sec a x + \tan a x} } + C$
The result follows on setting $a = 1$.
$\blacksquare$
Proof 2
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) | \(=\) | \(\ds \sec a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds a \sec a x \tan a x\) | Derivative of Function of Constant Multiple, Derivative of $\sec$ |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \sec^2 x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \tan x\) | Primitive of $\sec^2 x$ |
Then:
\(\ds \int \sec^3 x \rd x\) | \(=\) | \(\ds \int \sec x \sec^2 x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sec x \tan x - \int \tan x \paren {\sec x \tan x} \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \sec x \tan x - \int \tan^2 x \sec x \rd x + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \sec x \tan x - \int \paren {\sec^2 x - 1} \sec x \rd x + C\) | Difference of $\sec^2$ and $\tan^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sec x \tan x - \int \sec^3 x \rd x + \int \sec x \rd x + C\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \sec x \tan x - \int \sec^3 x \rd x + \ln \size {\sec x + \tan x} + C\) | Primitive of $\sec x$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \int \sec^3 a \rd x\) | \(=\) | \(\ds \sec x \tan x + \ln \size {\sec x + \tan x} + C\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \sec^3 a \rd x\) | \(=\) | \(\ds \dfrac 1 2 \paren {\sec x \tan x + \ln \size {\sec a x + \tan a x} } + C\) |
$\blacksquare$