Primitive of Sine of a x squared by Cosine of a x squared
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Theorem
- $\ds \int \sin^2 a x \cos^2 a x \rd x = \frac x 8 - \frac {\sin 4 a x} {32 a} + C$
Proof 1
\(\ds \int \sin^2 a x \cos^2 a x \rd x\) | \(=\) | \(\ds \int \sin^2 a x \paren {1 - \sin^2 a x} \rd x\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \sin^2 a x \rd x - \int \sin^4 a x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 2 - \frac {\sin 2 a x} {4 a} - \int \sin^4 a x \rd x + C\) | Primitive of $\sin^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 2 - \frac {\sin 2 a x} {4 a} - \paren {\frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} } + C\) | Primitive of $\sin^4 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 8 - \frac {\sin 4 a x} {32 a} + C\) | gathering terms and simplifying |
$\blacksquare$
Proof 2
\(\ds \int \sin^2 a x \cos^2 a x \rd x\) | \(=\) | \(\ds \int \paren {\sin a x \cos a x}^2 \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {\frac 1 2 \sin 2 a x}^2 \rd x\) | Double Angle Formula for Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \int \sin^2 2 a x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \paren {\frac x 2 - \frac {\map \sin {2 x \times 2 a} } {4 \times 2 a} } + C\) | Primitive of $\sin^2 a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac x 8 - \frac {\sin 4 a x} {32 a} + C\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.403$