Primitive of Reciprocal of Sine of a x minus Cosine of a x
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Contents
Theorem
- $\displaystyle \int \frac {\mathrm d x} {\sin a x - \cos a x} = \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac {a x} 2 - \frac \pi 8}\right)}\right\vert + C$
Proof
\(\displaystyle \int \frac {\mathrm d x} {\sin a x - \cos a x}\) | \(=\) | \(\displaystyle \int \frac {\mathrm d x} {\sqrt 2 \cos \left({a x - \dfrac {3 \pi} 4}\right)}\) | Sine of x minus Cosine of x: Cosine Form | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {\sqrt 2} \int \sec \left({a x - \dfrac {3 \pi} 4}\right) \ \mathrm d x\) | Secant is Reciprocal of Cosine |
Let:
\(\displaystyle z\) | \(=\) | \(\displaystyle a x - \dfrac {3 \pi} 4\) | |||||||||||
\(\displaystyle \implies \ \ \) | \(\displaystyle \frac {\mathrm d z} {\mathrm d x}\) | \(=\) | \(\displaystyle a\) | Derivative of Identity Function and Derivatives of Function of $a x + b$ |
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\(\displaystyle \implies \ \ \) | \(\displaystyle \frac 1 {\sqrt 2} \int \sec \left({a x - \dfrac {3 \pi} 4}\right) \ \mathrm d x\) | \(=\) | \(\displaystyle \frac 1 {a \sqrt 2} \int \sec z \ \mathrm d z\) | Integration by Substitution | |||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac z 2 + \frac \pi 4}\right)}\right\vert + C\) | Primitive of $\sec a x$ | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac 1 2 \left({a x - \dfrac {3 \pi} 4}\right) + \frac \pi 4}\right)}\right\vert + C\) | substituting for $z$ | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {a \sqrt 2} \ln \left\vert{\tan \left({\frac {a x} 2 - \frac \pi 8}\right)}\right\vert + C\) | 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 $\cos a x$: $14.412$