Question
the integral from 0 to pi/2 of sin^5(x)cos^18(x)
Answers
MathMate
When it is a product of odd powers of sin/cos, write it as a product of an even power, and rewrite dx as cosxdx, or d(sin(x)), after which it will be a simple substitution.
For example,
∫sin^2(x) cos^3(x)dx
=∫sin^2(x) (1-cos^2(x) dsin(x)
=∫y²(1-y²)dy
=y³/3-y<sup>5</sup>/5+C
=sin³(x)/3 - sin<sup>5</sup>(x)/5 + C
For example,
∫sin^2(x) cos^3(x)dx
=∫sin^2(x) (1-cos^2(x) dsin(x)
=∫y²(1-y²)dy
=y³/3-y<sup>5</sup>/5+C
=sin³(x)/3 - sin<sup>5</sup>(x)/5 + C