Math CentralQuandaries & Queries


Question from ascher, a student:

how do you integrate this equation
∫ sin^3 (2x) dx

Hi Ascher,

Write sin3(2x) as sin2(2x) sin(2x) and then use the fact that sin2(y) + cos2(y) = 1.

Write back if you need more help,



Remember how recipes say "set one egg aside"? Well, set one of the sines aside:

integral of sin2 (2x) sin(2x) dx

Now use the Pythagorean identity to rewrite in cosines:

integral of (1 - cos2(2x)) sin(2x) dx

and integrate by substitution, using u = cos(2x), (-1/2)du = sin(x)dx. This or a similar trick can be used whenever you have:

-an odd power of sine and any power of cosine
-an odd power of cosine and any power of sine
-an even power of secant and any power of tangent
-an odd power of tangent and any power of secant

Only the cases with sineven coseven and taneven secodd need special methods.

Good Hunting!

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