Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x

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Theorem

$\ds \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$


Proof

Let:

\(\ds u\) \(=\) \(\ds \sin a x\)
\(\ds \frac {\d u} {\d x}\) \(=\) \(\ds a \cos a x\) Derivative of $\sin a x$


Then:

\(\ds \int \frac {\d x} {\cos a x \paren {1 - \sin a x} }\) \(=\) \(\ds \int \frac {\cos a x \rd x} {\cos^2 a x \paren {1 - \sin a x} }\) multiplying top and bottom by $\cos a x$
\(\ds \) \(=\) \(\ds \int \frac {\cos a x \rd x} {\paren {1 - \sin^2 a x} \paren {1 - \sin a x} }\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\d u} {\paren {1 - u^2} \paren {1 - u} }\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\d u} {\paren {1 + u} \paren {1 - u}^2}\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \frac 1 a \paren {\frac 1 2 \paren {\frac 1 {1 - u} + \frac 1 2 \ln \size {\frac {1 + u} {1 - u} } } } + C\) Primitive of $\dfrac 1 {\paren {a x + b}^2 \paren {p x + q} }$
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - u} } + \frac 1 {4 a} \ln \size {\frac {1 + u} {1 - u} } + C\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {1 + \sin a x} {1 - \sin a x} } + C\) substituting for $u$
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\frac 1 2 \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} } {\frac 1 2 \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} } } + C\) Reciprocal of $1 - \sin$ and Reciprocal of $1 + \sin$
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\map {\cos^2} {\frac \pi 4 - \frac {a x} 2} } {\map {\cos^2} {\frac \pi 4 + \frac {a x} 2} } } + C\) Secant is Reciprocal of Cosine
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\map {\sin^2} {\frac \pi 2 - \paren {\frac \pi 4 - \frac {a x} 2} } } {\map {\cos^2} {\frac \pi 4 + \frac {a x} 2} } } + C\) Sine of Complement equals Cosine
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\map {\sin^2} {\frac \pi 4 + \frac {a x} 2} } {\map {\cos^2} {\frac \pi 4 + \frac {a x} 2} } } + C\)
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\map {\tan^2} {\frac \pi 4 + \frac {a x} 2} } + C\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C\) Logarithm of Power

$\blacksquare$


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