Primitive of Sine Function/Corollary
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Corollary to Primitive of Sine Function
- $\ds \int \sin a x \rd x = - \frac {\cos a x} a + C$
where $a$ is a non-zero constant.
Proof
| \(\ds \int \sin x \rd x\) | \(=\) | \(\ds -\cos x + C\) | Primitive of $\sin x$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \sin a x \rd x\) | \(=\) | \(\ds \frac 1 a \paren {-\cos a x} + C\) | Primitive of Function of Constant Multiple | ||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac {\cos a x} a + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\cos a x$
- Primitive of $\tan a x$
- Primitive of $\cot a x$
- Primitive of $\sec a x$
- Primitive of $\csc a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$: $14.339$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $56$.
- 1983: K.G. Binmore: Calculus ... (previous) ... (next): $9$ Sums and Integrals: $9.8$ Standard Integrals