Primitive of Reciprocal of Sine of a x plus Cosine of a x/Proof 2

Theorem

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

Proof

\(\ds \int \frac {\d x} {\sin a x + \cos a x}\)

\(=\)

\(\ds \frac 1 a \int \frac {\dfrac {2 \rd u} {1 + u^2} } {\dfrac {2 u} {1 + u^2} + \dfrac {1 - u^2} {1 + u^2} }\)

Weierstrass Substitution: $u = \tan \dfrac {a x} 2$

\(\ds \)

\(=\)

\(\ds \frac 2 a \int \frac {\d u} {- u^2 + 2 u + 1}\)

simplifying

\(\ds \)

\(=\)

\(\ds \frac 2 a \paren {\frac 1 {\sqrt 8} \ln \size {\frac {-2 u + 2 - \sqrt 8} {-2 u + 2 + \sqrt 8} } } + C\)

Primitive of $\dfrac 1 {a x^2 + b x + c}$

\(\ds \)

\(=\)

\(\ds \frac 1 {a \sqrt 2} \ln \size {\frac {u - 1 + \sqrt 2} {u - 1 - \sqrt 2} } + C\)

simplifying

\(\ds \)

\(=\)

\(\ds \frac 1 {a \sqrt 2} \ln \size {\frac {\tan \dfrac {a x} 2 - \paren {1 - \sqrt 2} } {\tan \dfrac {a x} 2 - \paren {1 + \sqrt 2} } } + C\)

substituting for $u$

\(\ds \)

\(=\)

\(\ds \frac 1 {a \sqrt 2} \ln \size {\frac {\tan \dfrac {a x} 2 - \tan \dfrac \pi 8} {\tan \dfrac {a x} 2 - \tan \dfrac {3 \pi} 8} } + C\)

Tangent of $\dfrac \pi 8$ and Tangent of $\dfrac {3 \pi} 8$

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