the integral from 0 to pi/4 of (cot2theta)^1/3
This is suppose to somehow transform into the form of cos^(a)theta.sin^(b)theta
where a and b are constants
but i've tried so many times to put it in this form but the 1/3 is just making things difficult for me
The definition ofOriginally Posted by CookieC
is
, the limits being from t=0 to t=1. The given integral goes from
to
. So you need a change of variable which achieves that change of limits. I suggest making the substitution
, which has that effect. Then
, and the integral becomes
)^{1/3}}{(\sin(2\theta))^{1/3}}d\theta &= \int_0^1\frac{(1-t)^{1/6}}{t^{1/6}}\,\frac{dt}{4t^{1/2}(1-t)^{1/2}} \\ &= \frac14\int_0^1t^{-2/3}(1-t)^{-1/3}\,dt \\ &=\tfrac14 B\bigl(\tfrac13,\tfrac23\bigr).\end{aligned})
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