\[\int \sin^{2}(2x)\cos^{3}(2x) \, dx\]

1
Remove parentheses.
sin2(2x)cos3(2x)dx\int \sin^{2}(2x)\cos^{3}(2x) \, dx

2
Use Pythagorean Identities: sin2x+cos2x=1\sin^{2}x+\cos^{2}x=1.
sin2(2x)(1sin2(2x))cos2xdx\int \sin^{2}(2x)(1-\sin^{2}(2x))\cos{2x} \, dx

3
Use Integration by Substitution.
Let u=sin2xu=\sin{2x}, du=2cos2xdxdu=2\cos{2x} \, dx, then cos2xdx=12du\cos{2x} \, dx=\frac{1}{2} \, du

4
Using uu and dudu above, rewrite sin2(2x)(1sin2(2x))cos2xdx\int \sin^{2}(2x)(1-\sin^{2}(2x))\cos{2x} \, dx.
u2(1u2)2du\int \frac{{u}^{2}(1-{u}^{2})}{2} \, du

5
Use Constant Factor Rule: cf(x)dx=cf(x)dx\int cf(x) \, dx=c\int f(x) \, dx.
12u2(1u2)du\frac{1}{2}\int {u}^{2}(1-{u}^{2}) \, du

6
Expand.
12u2u4du\frac{1}{2}\int {u}^{2}-{u}^{4} \, du

7
Use Power Rule: xndx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
u36u510\frac{{u}^{3}}{6}-\frac{{u}^{5}}{10}

8
Substitute u=sin2xu=\sin{2x} back into the original integral.
sin3(2x)6sin5(2x)10\frac{\sin^{3}(2x)}{6}-\frac{\sin^{5}(2x)}{10}

9
Add constant.
sin3(2x)6sin5(2x)10+C\frac{\sin^{3}(2x)}{6}-\frac{\sin^{5}(2x)}{10}+C

Done

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