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I am new to calculus and am looking for some feedback regarding the following question. Many thanks in advance!

The following line integral is given: $\int_C(y + y cos(xy))dx + (x + x cos(xy))dy$. Which of the following statements are correct?

  1. If $C=(x-3)^2+(y-4)^2=1$, with clockwise movement, then the integral is 0.

  2. If C is circular arc $x^2+y^2=8$ from point (2,2) to point (-2,-2), then the integral is not 0.

  3. If $1 \le t \le\frac{π}{2}$ and $C=\mathbf C(t)=t\mathbf i+\frac{π}{t}\mathbf j$, then the integral is 0.

My interpretation is the following:

I found the potential φ: yx+sin(xy), which means that the outcome of the integral will be independent of its path, and can be calculated via $φ(x_1,y_1)-φ(x_0,y_0)$.

This yields the following results:

  1. always 0
  2. 0
  3. not 0, based on $φ(\frac{π}{2},2)-φ(1,π)$.

Thus 1 is correct and 2 and 3 are incorrect.

Any comments will be very much appreciated!

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  • $\begingroup$ i think you didnt put spaces the right way in your integral $\endgroup$ – Milan May 18 at 18:12
  • $\begingroup$ for some reason it will not let me separate the expression any more than my edit... $\endgroup$ – dalta May 18 at 18:15
  • $\begingroup$ sorry, i meant brackets not spaces haha $\endgroup$ – Milan May 18 at 18:17
  • $\begingroup$ you were right! $\endgroup$ – dalta May 18 at 18:24

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