Primitive of Reciprocal of 1 plus Sine of a x
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Theorem
- $\ds \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C$
Corollary
- $\ds \int \frac {\d x} {1 + \sin x} = \map \tan {\frac x 2 - \frac \pi 4} + C$
Proof
\(\ds \int \frac {\d x} {1 + \sin a x}\) | \(=\) | \(\ds \int \frac 1 2 \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x\) | Reciprocal of One Plus Sine | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac 1 2 \int \map {\sec^2} {\frac \pi 4 - \frac {a x} 2} \rd x\) | Primitive of Constant Multiple of Function |
Then:
\(\ds z\) | \(=\) | \(\ds \frac \pi 4 - \frac {a x} 2\) | ||||||||||||
\(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds -\frac a 2\) | Derivative of Power | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {1 + \sin a x}\) | \(=\) | \(\ds -\frac 1 2 \int \frac 2 a \sec^2 z \rd z\) | Integration by Substitution from $(1)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 2 \cdot \frac 2 a \tan z + C\) | Primitive of Square of Secant Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C\) | substituting for $z$ |
$\blacksquare$
Also see
- Primitive of $\dfrac 1 {1 - \sin a x}$
- Primitive of $\dfrac 1 {1 + \cos a x}$
- Primitive of $\dfrac 1 {1 - \cos a x}$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.356$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $72$.