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