Primitive of Reciprocal of Power of Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\cos^n a x} = \frac {\sin a x} {a \paren {n - 1} \cos^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x}$
Proof
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:
\(\ds u\) | \(=\) | \(\ds \frac 1 {\cos^{n - 2} a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos^{- n + 2} a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -a \paren {-n + 2} \cos^{-n + 1} a x \sin a x\) | Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {a \paren {n - 2} \sin a x} {\cos^{n - 1} a x}\) | simplifying |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \frac 1 {\cos^2 a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sec^2 a x\) | Secant is $\dfrac 1 \cos$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {\tan a x} a\) | Primitive of $\sec^2 a x$ |
Then:
\(\ds \int \frac {\d x} {\cos^n a x}\) | \(=\) | \(\ds \int \frac {\d x} {\cos^{n - 2} a x \cos^2 a x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac 1 {\cos^{n - 2} a x} } \paren {\frac {\tan a x} a} - \int \paren {\frac {\tan a x} a} \paren {\frac {a \paren {n - 2} \sin a x} {\cos^{n - 1} a x} } \rd x\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin a x} {a \cos^{n - 1} a x} - \int \frac {\paren {n - 2} \sin^2 a x} {\cos^n a x} \rd x\) | Tangent is $\dfrac {\sin} {\cos}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin a x} {a \cos^{n - 1} a x} - \int \frac {\paren {n - 2} \paren {1 - \cos^2 a x} } {\cos^n a x} \rd x\) | Sum of $\sin^2$ and $\cos^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin a x} {a \cos^{n - 1} a x} - \paren {n - 2} \int \frac {\d x} {\cos^n a x} + \paren {n - 2} \int \frac {\d x} {\cos^{n - 2} a x}\) | Linear Combination of Primitives | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {n - 1} \int \frac {\d x} {\cos^n a x}\) | \(=\) | \(\ds \frac {\sin a x} {a \cos^{n - 1} a x} + \paren {n - 2} \int \frac {\d x} {\cos^{n - 2} a x}\) | gathering terms | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {\cos^n a x}\) | \(=\) | \(\ds \frac {\sin a x} {a \paren {n - 1} \cos^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x}\) | dividing by $n - 1$ |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.397$