Primitive of Reciprocal of Sine of a x minus Cosine of a x
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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
\(\ds \int \frac {\d x} {\sin a x - \cos a x}\) | \(=\) | \(\ds \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac {3 \pi} 4} }\) | Sine of x minus Cosine of x: Cosine Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac {3 \pi} 4} \rd x\) | Secant is Reciprocal of Cosine |
Let:
\(\ds z\) | \(=\) | \(\ds a x - \dfrac {3 \pi} 4\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds a\) | Derivative of Identity Function and Derivatives of Function of $a x + b$ |
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\(\ds \leadsto \ \ \) | \(\ds \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac {3 \pi} 4} \rd x\) | \(=\) | \(\ds \frac 1 {a \sqrt 2} \int \sec z \rd z\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac z 2 + \frac \pi 4} } + C\) | Primitive of $\sec a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac 1 2 \paren {a x - \dfrac {3 \pi} 4} + \frac \pi 4} } + C\) | substituting for $z$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 - \frac \pi 8} } + C\) | simplifying |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.412$