Primitive of Reciprocal of Sine of a x plus Cosine of a x

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Theorem

$\displaystyle \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$


Proof 1

\(\displaystyle \int \frac {\d x} {\sin a x + \cos a x}\) \(=\) \(\displaystyle \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac \pi 4} }\) Sine of x plus Cosine of x: Cosine Form
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x\) Secant is Reciprocal of Cosine


Let:

\(\displaystyle z\) \(=\) \(\displaystyle a x - \dfrac \pi 4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d z} {\d x}\) \(=\) \(\displaystyle a\) Derivative of Identity Function
and Derivatives of Function of $a x + b$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x\) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \int \sec z \rd z\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac z 2 + \frac \pi 4} } + C\) Primitive of $\sec a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac 1 2 \paren {a x - \dfrac \pi 4} + \frac \pi 4} } + C\) substituting for $z$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C\) simplifying

$\blacksquare$


Proof 2

\(\displaystyle \int \frac {\d x} {\sin a x + \cos a x}\) \(=\) \(\displaystyle \frac 1 a \int \frac {\dfrac {2 \rd u} {1 + u^2} } {\dfrac {2 u} {1 + u^2} + \dfrac {1 - u^2} {1 + u^2} }\) Weierstrass Substitution: $u = \tan \dfrac {a x} 2$
\(\displaystyle \) \(=\) \(\displaystyle \frac 2 a \int \frac {\d u} {- u^2 + 2 u + 1}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 2 a \paren {\frac 1 {\sqrt 8} \ln \size {\frac {-2 u + 2 - \sqrt 8} {-2 u + 2 + \sqrt 8} } } + C\) Primitive of $\dfrac 1 {a x^2 + b x + c}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \size {\frac {u - 1 + \sqrt 2} {u - 1 - \sqrt 2} } + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \size {\frac {\tan \dfrac {a x} 2 - \paren {1 - \sqrt 2} } {\tan \dfrac {a x} 2 - \paren {1 + \sqrt 2} } } + C\) substituting for $u$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a \sqrt 2} \ln \size {\frac {\tan \dfrac {a x} 2 - \tan \dfrac \pi 8} {\tan \dfrac {a x} 2 - \tan \dfrac {3 \pi} 8} } + C\) Tangent of $\dfrac \pi 8$ and Tangent of $\dfrac {3 \pi} 8$


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