Primitive of Sine of a x by Cosine of a x
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Theorem
- $\ds \int \sin a x \cos a x \rd x = \frac {\sin^2 a x} {2 a} + C$
Proof
| \(\ds \int \sin a x \cos a x \rd x\) | \(=\) | \(\ds \int \frac {\sin 2 a x} 2 \rd x\) | Double Angle Formula for Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\cos 2 a x} {4 a} + C\) | Primitive of $\cos a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\paren {1 - 2 \sin^2 a x} } {4 a} + C\) | Double Angle Formula for Cosine: Corollary $2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-1} {4 a} + \frac {\sin^2 a x} {2 a} + C\) | separating fraction | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin^2 a x} {2 a} + C\) | subsuming $\dfrac {-1} {4 a}$ into arbitrary constant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.399$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $63$.
- 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: $(19)$ Integrals Involving $\sin a x$ and $\cos a x$: $17.19.1.$