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

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

$\ds \int \sin^2 x \cos^3 x \rd x = \dfrac {\sin^3 x} 3 - \dfrac {\sin^5 x} 5 + C$

Proof

\(\ds \int \sin^2 x \cos^3 x \rd x\)

\(=\)

\(\ds \int \sin^2 x \cos^2 x \cos x \rd x\)

\(\ds \)

\(=\)

\(\ds \int \sin^2 x \paren {1 - \sin^2 x} \sin^2 x \cos x \rd x\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds \int \paren {\sin^2 x - \sin^4 x} \cos x \rd x\)

\(\ds \)

\(=\)

\(\ds \int \sin^2 x \cos x \rd x - \int \sin^4 x \cos x \rd x\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \dfrac {\sin^3 x} 3 - \dfrac {\sin^5 x} 5 + C\)

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: Exercises $\text {XIV}$: $11$.