How to integrate this trigonometric integral?

Question

I'm currently quite stumped by an integration problem involving a sine to the second degree. The problem goes as follows: $$\int(8.89412\sin(0.047169x+0.105306)+21.8786)^2~dx$$ I understand that I must expand this expression in order to get further and I also understand that I must later use a reduction formula in order to reduce the exponent of sine from $\sin^2$ to just $\sin$. However, I unfortunately do not fully understand the integration process involving the reduction formula. It would be incredibly helpful if anybody could outline the full integration process. I think having that would explain it to me much better. Also, I have not been able to find a satisfactory explanation for the derivation of the reduction formula so having an explanation for how you can get the reduction formula for $\sin^2$ would be incredibly helpful as well. Thank you so much.

P.S sorry for the poor MathJax, this is my first time using it properly.

Answer 1

Hint: $\displaystyle\sin^2(x)=\frac{\sin^2(x)+\cos^2(x)+\sin^2(x)-\cos^2(x)}2=\frac{1-\cos(2x)}2.$

Answer 2

When you expand the term in the middle out, you're going to get something of the form

$\alpha_1 \sin^2(kx + \theta) + \alpha_2 \sin(kx + \theta) + \alpha_3$

So I'm assuming you can already handle the last two terms, and you're mostly stuck on the first one (in case you need a hint, the integral of $\sin(x)$ is $-\cos(x)$).

What you need to do for the first term is use a double angle identity. In particular, we use the identity $\cos(2\alpha) = 1 - 2 \sin^2(\alpha)$. Or, when rearranged, $\sin^2(\alpha) = \frac{1}{2}(1 - \cos(2\alpha))$. Putting that into the integral, you'll now have something of the form

$\beta_1 \cos(Kx + \Theta) + \beta_2 \sin(kx + \theta) + \beta_3$

where the various parameters have been appropriately identified, and these are all fairly simple terms to integrate.