Integral of sin(x) sin(x+1) dx from 0 to 2pi. This is ##\sin(x- (x+1))##, or ##\sin(1)##. So now we have the even more nonsensical ##\sin(1)=0##, which obviously is not true. What happened is that you dropped a constant of integration. That supposedly nonsensical ##\sin(1)=0## is better stated as ##\sin(1) = c##. Instead of reaching a contradiction you've reached a tautology.
Bottom line: That's Just one of those things you need to be aware of when you do integration by parts. Here's a hint that will help you solve this integral: Use the trick that x- x=0. With this, cos(a) = cos(a+x- x) = cos((a+x)- x) = cos(a+x)cos(x) + sin(a+x)sin(x).
Now substitute a=1.
Solve the integral = - ln . Alternate Form of Result.
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