I do NOT get the sum of four terms, I get the sum of three terms:
On the path x= sin t, , dx= cos t dt and .
and
The integral becomes .
Use integration by parts to do the first integral.
If , find
where is the directed path from (0,0) to (1,1) along the graph of the vector equation:
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Here's what I did:
I recognized that the integral is not independent of path... So set up:
which gives me the sum of 4 integral terms... I evaluated these up to the following step so somebody could double check:
Now does this answer seem correct to you?
I do NOT get the sum of four terms, I get the sum of three terms:
On the path x= sin t, , dx= cos t dt and .
and
The integral becomes .
Use integration by parts to do the first integral.
Yes, there were a couple of typos in HoI's reply which you have spotted and fixed.
See integrate x (Sin[x])^2 Cos[x] - Wolfram|Alpha and click on Show steps.