(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

To show that [tex]\int_{0}^ \frac{\pi}{2}\sqrt{cos\theta}sin^3(\theta) d\theta[/tex] = 8/21

3. The attempt at a solution

The above expression was simplified as

[tex]\int_{0}^ \frac{\pi}{2}\sqrt{cos\theta}sin^2(\theta) sin(\theta) d\theta[/tex]

[tex]\int_{0}^ \frac{\pi}{2}\sqrt{cos\theta}(1-cos^2(\theta)) sin\theta d\theta[/tex]

I have tried using integration by substitution method.

Let [tex]cos\theta = t^2[/tex]

or,[tex] sin\theta d\theta = 2tdt[/tex]

also changing the limits, when [tex]\theta = 0[/tex] , t becomes 1

and when [tex]\theta = \frac{\pi}{2} [/tex], t becomes 0

therefore the expression will look like this.

[tex]\int_{1}^ 0 t.(t^4-1)2t.dt[/tex]

Am I going in to the right direction or should I use any other method like integration by parts.

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# Integration by substitution of sqrt cos theta.sin cube theta

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