Primitive of Sine Function/Corollary

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.

$\ds \int \sin x \rd x$

$=$

$\ds -\cos x + C$

$\ds \leadsto \ \ $

$\ds \int \sin a x \rd x$

$=$

$\ds \frac 1 a \paren {-\cos a x} + C$

$\ds $

$=$

$\ds -\frac {\cos a x} a + C$

simplifying

$\blacksquare$

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
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(17)$ Integrals Involving $\sin a x$: $17.17.1.$