# Thread: Integration of sin to the power of a fration

1. ## Integration of sin to the power of a fration

Hi, I've spent the best part of the day trying to get beyond this point and my trawling searches of the internet have sadly turned up no answers.

I am trying to integrate the equation shown in the picture above, I thought about converting the sin^2/3 to its trigonometric identity, but the only examples I can find are for when power is 2 or more. Anyone got any ideas where I could look for inspiration? or better yet able to offer an explanation. This day has killed me

2. Originally Posted by mrlibertine
Hi, I've spent the best part of the day trying to get beyond this point and my trawling searches of the internet have sadly turned up no answers.

I am trying to integrate the equation shown in the picture above, I thought about converting the sin^2/3 to its trigonometric identity, but the only examples I can find are for when power is 2 or more. Anyone got any ideas where I could look for inspiration? or better yet able to offer an explanation. This day has killed me
Where has this integral come from? It has no closed form using elementary functions but can be found using the hypergeometric function.

Perhaps there's a typo and it's meant to be $\displaystyle \int \sin^{2/3} (3t) \, {\color{red}\cos (3t)} \, dt$.

3. Its part of my workings out so far.

at the risk of humiliating myself I attach what I have 'achieved' so far.

4. Originally Posted by mrlibertine
Its part of my workings out so far.

at the risk of humiliating myself I attach what I have 'achieved' so far.

Your mistake is in saying that $\displaystyle \int \cot (3t) \, dt = |\sin (3t)|$. It's not. $\displaystyle \int \cot (3t) \, dt = \frac{1}{3} |\sin (3t)|$. Therefore the integrating factor is $\displaystyle \sin^{1/3} (3t)$ NOT $\displaystyle \sin (3t)$. This makes a very big difference.

5. that makes sense, I just used a table conversion without thinking what I was doing. Thank you for your help

### cot3t identuty

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