Primitive of Power of Sine of a x by Power of Cosine of a x/Examples/cos 3rd x sin 4th x
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Example of Use of Primitive of $\sin^m x \cos^n x$
- $\ds \int \cos^3 x \sin^4 x \rd x = \dfrac {\sin^5 x} 5 - \dfrac {\sin^7 x} 7 + C$
Proof
| \(\ds \int \cos^3 x \sin^4 x \rd x\) | \(=\) | \(\ds \int \cos^2 x \sin^4 x \cos x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \paren {1 - \sin^2 x} \sin^4 x \cos x \rd x\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \paren {\sin^4 x - \sin^6 x} \cos x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \sin^4 x \cos x \rd x - \int \sin^6 x \cos x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sin^5 x} 5 - \dfrac {\sin^7 x} 7 + C\) | Primitive of Power of $\sin a x$ by $\cos a x$ |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Algebraic Integration: Powers of cos and sine: Example 1.