So i understand this question mostly, i dont understand where they got the (1+4) from? and shouldnt there be a negative in front of the last part on the 2nd line.
At the end of the second line what you have is:
Notice that the last term is common with the left hand side, so move it to the other side to get:
And no, the plus sign before the last term of the 2nd line is correct. Starting with
and using integration by parts, let , ,
I agree, that is hard to make sense of- they leave a lot out. The problem is to integrate . They note that "it doesn't matter which function is chosen to differentiate" and then choose to differentiate the .
That is we let and . Then and .
, then, is
Of course, so that first term is 0. That's the first "0" in the second line. What's left is .
Now, do the integration by parts again, letting and so that and .
so the first term is
The integral, at this point is
Now, here's the point. Because the first integral takes us from "cos(t)" to "sin(t)" and the second integral takes back to "cos(t)", while the derivatives of always gives " ", we have come right back (that integral on the right) to what we started with (the integral on the left). So combine those by adding , giving on the left. That is where the "4+ 1= 5" on the left came from.
same______integrate **************(line one)
-(diff________same) *******************(line 2)
This is a boss way for noobs like me to remember the order. brackets mean with an integral sign at the front.