Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x
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Theorem
- $\ds \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$
Proof
Let:
\(\ds u\) | \(=\) | \(\ds \sin a x\) | ||||||||||||
\(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds a \cos a x\) | Derivative of $\sin a x$ |
Then:
\(\ds \int \frac {\d x} {\cos a x \paren {1 - \sin a x} }\) | \(=\) | \(\ds \int \frac {\cos a x \rd x} {\cos^2 a x \paren {1 - \sin a x} }\) | multiplying top and bottom by $\cos a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos a x \rd x} {\paren {1 - \sin^2 a x} \paren {1 - \sin a x} }\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d u} {\paren {1 - u^2} \paren {1 - u} }\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {\d u} {\paren {1 + u} \paren {1 - u}^2}\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \paren {\frac 1 2 \paren {\frac 1 {1 - u} + \frac 1 2 \ln \size {\frac {1 + u} {1 - u} } } } + C\) | Primitive of $\dfrac 1 {\paren {a x + b}^2 \paren {p x + q} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - u} } + \frac 1 {4 a} \ln \size {\frac {1 + u} {1 - u} } + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {1 + \sin a x} {1 - \sin a x} } + C\) | substituting for $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\frac 1 2 \map {\sec^2} {\frac \pi 4 + \frac {a x} 2} } {\frac 1 2 \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} } } + C\) | Reciprocal of $1 - \sin$ and Reciprocal of $1 + \sin$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\map {\cos^2} {\frac \pi 4 - \frac {a x} 2} } {\map {\cos^2} {\frac \pi 4 + \frac {a x} 2} } } + C\) | Secant is Reciprocal of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\map {\sin^2} {\frac \pi 2 - \paren {\frac \pi 4 - \frac {a x} 2} } } {\map {\cos^2} {\frac \pi 4 + \frac {a x} 2} } } + C\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\frac {\map {\sin^2} {\frac \pi 4 + \frac {a x} 2} } {\map {\cos^2} {\frac \pi 4 + \frac {a x} 2} } } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {4 a} \ln \size {\map {\tan^2} {\frac \pi 4 + \frac {a x} 2} } + C\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C\) | Logarithm of Power |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {\sin a x \paren {1 + \cos a x} }$
- Primitive of $\dfrac 1 {\sin a x \paren {1 - \cos a x} }$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.410$