integration - How can I solve this trigonometric integral step by step?

I have already posted a question similar to this one so I'm sorry if you're seeing this again. However, I'd like to whether I could get help solving the below definite integral step by step because I do not understand how to get to the answer which my GDC calculator gives me. The problem involves a trigonometric integral with a sine to the second degree. The problem goes as follows:

$$\pi\int_0^{109.620851326}(8.89412\sin(0.047169x+0.105306)+21.8786)^2~dx$$
My attempts at the solution to both the indefinite and definite integral involved the expression for the reduction of $\sin^2$ which is the correct approach. I reduced sine in the following manner:
$$\sin^2\theta=\frac{1-\cos{2\theta}}{2}$$
The answer to the definite integral also seems to be 15086.383263155 so the goal is to reach that answer. Whatever I seem to do I always get around 642.78185152953. I know that it will be difficult to explain every step and for that I apologise, I just haven't been able to find help to do this anywhere else. I really appreciate any help! :)
Thank you.

P.S Sorry for the poor MathJax, I'm new to the system .