Primitive of Power of Cosecant of a x
Jump to navigation
Jump to search
Theorem
- $\ds \int \csc^n a x \rd x = \frac{-\csc^{n - 2} a x \cot a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \csc^{n - 2} a x \rd x$
where $n \ne -1$.
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:
\(\ds u\) | \(=\) | \(\ds \csc^{n - 2} a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -a \paren {n - 2} \csc^{n - 3} a x \csc a x \cot a x\) | Derivative of Power, Derivative of $\csc$, Chain Rule for Derivatives | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -a \paren {n - 2} \csc^{n - 2} a x \cot a x\) |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \csc^2 a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {-\cot a x} a\) | Primitive of $\csc^2 a x$ |
Then:
\(\ds \int \csc^n a x \rd x\) | \(=\) | \(\ds \int \csc^{n - 2} a x \csc^2 a x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \csc^{n - 2} a x \paren {\frac {-\cot a x} a} - \int \paren {\frac {-\cot a x} a} \paren {-a \paren {n - 2} \csc^{n - 2} a x \cot a x } \rd x\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\csc^{n - 2} a x \cot a x} a - \paren {n - 2} \int \cot^2 a x \csc^{n - 2} a x \rd x\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\csc^{n - 2} a x \cot a x} a - \paren {n - 2} \int \paren {\csc^2 a x - 1} \csc^{n - 2} a x \rd x\) | Difference of $\csc^2$ and $\cot^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\csc^{n - 2} a x \cot a x} a - \paren {n - 2} \int \csc^n a x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {n - 2} \int \csc^{n - 2} a x \rd x\) | |||||||||||
\(\ds \paren {n - 1} \int \csc^n a x \rd x\) | \(=\) | \(\ds \frac {-\csc^{n - 2} a x \cot a x} a + \paren {n - 2} \int \csc^{n - 2} a x \rd x\) | gathering terms | |||||||||||
\(\ds \int \csc^n a x \rd x\) | \(=\) | \(\ds \frac{-\csc^{n - 2} a x \cot a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \csc^{n - 2} a x \rd x\) | dividing by $n - 1$ |
$\blacksquare$
Also see
- Primitive of $\csc a x$ for the case where $n = 1$
- Primitive of $\sin^n a x$
- Primitive of $\cos^n a x$
- Primitive of $\tan^n a x$
- Primitive of $\cot^n a x$
- Primitive of $\sec^n a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csc a x$: $14.470$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $93$.