Functions consisting of products of the sine and cosine can be integrated by using substitution and trigonometric identities. These can sometimes be tedious, but the technique is straightforward. Some examples will suffice to explain the approach.

Example 10.1.1 Evaluate $\ds\int \sin^5 x\,dx$. Rewrite the function: $$ \int \sin^5 x\,dx=\int \sin x \sin^4 x\,dx= \int \sin x (\sin^2 x)^2\,dx= \int \sin x (1-\cos^2 x)^2\,dx. $$ Now use $u=\cos x$, $du=-\sin x\,dx$: $$\eqalign{ \int \sin x (1-\cos^2 x)^2\,dx&=\int -(1-u^2)^2\,du\cr &=\int -(1-2u^2+u^4)\,du\cr &=-u+{2\over3}u^3-{1\over5}u^5+C\cr &=-\cos x+{2\over3}\cos^3 x-{1\over5}\cos^5x+C.\cr }$$

Example 10.1.2 Evaluate $\ds\int \sin^6 x\,dx$. Use $\ds \sin^2x =(1-\cos(2x))/2$ to rewrite the function: $$\eqalign{ \int \sin^6 x\,dx=\int (\sin^2 x)^3\,dx&= \int {(1-\cos 2x)^3\over 8}\,dx\cr &={1\over 8}\int 1-3\cos 2x+3\cos^2 2x-\cos^3 2x\,dx.\cr} $$ Now we have four integrals to evaluate: $$\int 1\,dx=x$$ and $$\int -3\cos 2x\,dx = -{3\over 2}\sin 2x$$ are easy. The $\ds \cos^3 2x$ integral is like the previous example: $$\eqalign{ \int -\cos^3 2x\,dx&=\int -\cos 2x\cos^2 2x\,dx\cr &=\int -\cos 2x(1-\sin^2 2x)\,dx\cr &=\int -{1\over 2}(1-u^2)\,du\cr &=-{1\over 2}\left(u-{u^3\over 3}\right)\cr &=-{1\over 2}\left(\sin 2x-{\sin^3 2x\over 3}\right).} $$ And finally we use another trigonometric identity, $\ds \cos^2x=(1+\cos(2x))/2$: $$ \int 3\cos^2 2x\,dx=3\int {1+\cos 4x\over 2}\,dx= {3\over 2}\left(x+{\sin 4x\over 4}\right). $$ So at long last we get $$ \int \sin^6 x\,dx = {x\over8} -{3\over 16}\sin 2x -{1\over 16}\left(\sin 2x-{\sin^3 2x\over 3}\right) +{3\over 16}\left(x+{\sin 4x\over 4}\right)+C. $$

Example 10.1.3 Evaluate $\ds\int\! \sin^2x\cos^2x\,dx$. Use the formulas $\ds \sin^2x =(1-\cos(2x))/2$ and $\ds \cos^2x =(1+\cos(2x))/2$ to get: $$ \int \sin^2x\cos^2x\,dx=\int {1-\cos(2x)\over2}\cdot {1+\cos(2x)\over2}\,dx. $$ The remainder is left as an exercise.

Exercises 10.1

Find the antiderivatives.

Ex 10.1.1 $\ds\int \sin^2 x\,dx$ (answer)

Ex 10.1.2 $\ds\int \sin^3 x\,dx$ (answer)

Ex 10.1.3 $\ds\int \sin^4 x\,dx$ (answer)

Ex 10.1.4 $\ds\int \cos^2 x\sin^3 x\,dx$ (answer)

Ex 10.1.5 $\ds\int \cos^3 x\,dx$ (answer)

Ex 10.1.6 $\ds\int \sin^2 x\cos^2 x\,dx$ (answer)

Ex 10.1.7 $\ds\int \cos^3 x \sin^2 x\,dx$ (answer)

Ex 10.1.8 $\ds\int \sin x (\cos x)^{3/2}\,dx$ (answer)

Ex 10.1.9 $\ds\int \sec^2 x\csc^2 x\,dx$ (answer)

Ex 10.1.10 $\ds\int \tan^3x \sec x\,dx$ (answer)