Primitive of Reciprocal of Sine of a x by 1 plus Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\sin a x \paren {1 + \cos a x} } = \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C$
Proof
Let:
\(\ds u\) | \(=\) | \(\ds \cos a x\) | ||||||||||||
\(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -a \sin a x\) | Derivative of $\cos a x$ |
Then:
\(\ds \int \frac {\d x} {\sin a x \paren {1 + \cos a x} }\) | \(=\) | \(\ds \int \frac {\sin a x \rd x} {\sin^2 a x \paren {1 + \cos a x} }\) | multiplying top and bottom by $\sin a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\sin a x \rd x} {\paren {1 - \cos^2 a x} \paren {1 + \cos a x} }\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {4 a} \ln \size {\frac {1 - \cos a x} {1 + \cos a x} } + C\) | substituting for $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {4 a} \ln \size {\frac {\frac 1 2 \sec^2 \frac {a x} 2} {\frac 1 2 \csc^2 \frac {a x} 2} } + C\) | Reciprocal of $1 - \cos$ and Reciprocal of $1 + \cos$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {4 a} \ln \size {\frac {\sin^2 \frac {a x} 2} {\cos^2 \frac {a x} 2} } + C\) | Secant is Reciprocal of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {4 a} \ln \size {\tan^2 \frac {a x} 2} + C\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a \paren {1 + \cos a x} } + \frac 1 {2 a} \ln \size {\tan \frac {a x} 2} + C\) | Logarithm of Power |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {\cos a x \paren {1 + \sin a x} }$
- Primitive of $\dfrac 1 {\cos a x \paren {1 - \sin a x} }$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.411$