# Question about line integrals with various options

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!

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