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

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Theorem

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


Proof

\(\ds \int \frac {\d x} {\paren {1 + \cos a x}^2}\) \(=\) \(\ds \int \paren {\frac 1 2 \sec^2 \frac {a x} 2}^2 \rd x\) Reciprocal of One Plus Cosine
\(\ds \) \(=\) \(\ds \frac 1 4 \int \sec^4 \frac {a x} 2 \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 4 \paren {\frac {\sec^2 \dfrac {a x} 2 \tan \dfrac {a x} 2} {\dfrac {3 a} 2} + \frac 2 3 \int \sec^2 \frac {a x} 2 \rd x} + C\) Primitive of $\sec^n a x$
\(\ds \) \(=\) \(\ds \frac 1 {6 a} \sec^2 \frac {a x} 2 \tan \dfrac {a x} 2 + \frac 1 6 \int \sec^2 \frac {a x} 2 \rd x + C\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {6 a} \sec^2 \frac {a x} 2 \tan \dfrac {a x} 2 + \frac 1 6 \paren {\frac 2 a \tan \frac {a x} 2} + C\) Primitive of $\sec^2 a x$
\(\ds \) \(=\) \(\ds \frac 1 {6 a} \paren {1 + \tan^2 \frac {a x} 2} \tan \dfrac {a x} 2 + \frac 2 {6 a} \tan \frac {a x} 2 + C\) Difference of Squares of Secant and Tangent
\(\ds \) \(=\) \(\ds \frac 1 {2 a} \tan \frac {a x} 2 + \frac 1 {6 a} \tan^3 \frac {a x} 2 + C\) simplifying

$\blacksquare$


Also see


Sources

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