Primitive of Power of Sine of a x by Power of Cosine of a x/Examples/cos squared x sin 4th x

Example of Use of Primitive of $\sin^m x \cos^n x$

$\ds \int \cos^2 x \sin^4 x \rd x = \dfrac 1 {32} \paren {2 x - \dfrac {\sin 2 x} 2 - \dfrac {2 \sin 4 x} 4 + \dfrac {\sin 6 x} 6} + C$

$\ds \int \cos^2 x \sin^4 x \rd x$

$=$

$\ds \int \paren {\cos^2 x \sin^2 x} \sin^2 x \rd x$

$\ds $

$=$

$\ds \int \paren {\dfrac {\sin^2 2 x} 4} \paren {\dfrac {1 - \cos 2 x} 2} \rd x$

$\ds $

$=$

$\ds \int \paren {\dfrac {1 - \cos 4 x} 8} \paren {\dfrac {1 - \cos 2 x} 2} \rd x$

$\ds $

$=$

$\ds \dfrac 1 {16} \int \paren {1 - \cos 2 x - \cos 4 x + \cos 2 x \cos 4 x} \rd x$

simplification

$\ds $

$=$

$\ds \dfrac 1 {16} \int \paren {1 - \cos 2 x - \cos 4 x + \dfrac {\cos 2 x + \cos 6 x} 2} \rd x$

some result

$\ds $

$=$

$\ds \dfrac 1 {32} \int \paren {2 - \cos 2 x - 2 \cos 4 x + \cos 6 x} \rd x$

simplification

$\ds $

$=$

$\ds \dfrac 1 {32} \paren {2 x - \dfrac {\sin 2 x} 2 - \dfrac {2 \sin 4 x} 4 + \dfrac {\sin 6 x} 6} + C$

various

$\blacksquare$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Algebraic Integration: Powers of cos and sine: Example