Primitive of Reciprocal of Square of 1 minus Cosine of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \frac {\d x} {\paren {1 - \cos a x}^2} = \frac {-1} {2 a} \cot \frac {a x} 2 - \frac 1 {6 a} \cot^3 \frac {a x} 2 + C$


Proof

\(\ds \int \frac {\d x} {\paren {1 - \cos a x}^2}\) \(=\) \(\ds \int \paren {\frac 1 2 \csc^2 \frac {a x} 2}^2 \rd x\) Reciprocal of One Minus Cosine
\(\ds \) \(=\) \(\ds \frac 1 4 \int \csc^4 \frac {a x} 2 \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 4 \paren {\frac{-\csc^2 \dfrac {a x} 2 \cot \dfrac {a x} 2} {\dfrac {3 a} 2} + \frac 2 3 \int \csc^2 \frac {a x} 2 \rd x} + C\) Primitive of $\csc^n a x$
\(\ds \) \(=\) \(\ds \frac {-1} {6 a} \csc^2 \frac {a x} 2 \cot \dfrac {a x} 2 + \frac 1 6 \int \csc^2 \frac {a x} 2 \rd x + C\) simplifying
\(\ds \) \(=\) \(\ds \frac {-1} {6 a} \csc^2 \frac {a x} 2 \cot \dfrac {a x} 2 + \frac 1 6 \paren {\frac {-2} a \cot \frac {a x} 2} + C\) Primitive of $\csc^2 a x$
\(\ds \) \(=\) \(\ds \frac {-1} {6 a} \paren {1 + \cot^2 \frac {a x} 2} \cot \dfrac {a x} 2 - \frac 2 {6 a} \cot \frac {a x} 2 + C\) Difference of Squares of Cosecant and Cotangent
\(\ds \) \(=\) \(\ds \frac {-1} {2 a} \cot \frac {a x} 2 - \frac 1 {6 a} \cot^3 \frac {a x} 2 + C\) simplifying

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_Reciprocal_of_Square_of_1_minus_Cosine_of_a_x&oldid=543973"