\( 2 = \int \frac { x ^ { 2 } d x } { 4 x ^ { 2 } + 1 } = \)

Determine whether or not the vector field \( (x-46 x y) i+\left(y^{2}-23 x^{2}\right) j \) is conservative. If it is, find a potential function. \( f(x, y)=\frac{x^{2}}{2}-23 x^{2} y+y^{3}+c \). The field is not conservative. \( f(x, y)=x^{2}-23 x^{2} y+\frac{y^{3}}{3}+c \). The field is not conservative. \( f(x, y)=\frac{x^{2}}{2}-x^{2} y+\frac{y^{3}}{3}+c \). The field is conservative. \( f(x, y)=\frac{x^{2}}{2}-23 x^{2} y+\frac{y^{3}}{3}+c \). The field is conservative. \( f \)

Compute the work done by the force field \( F \) along the curve \( C \). \( \mathrm{F}(x, y)=\langle 2 x, 2 y\rangle, C \) is the line segment from \( (1,1) \) to \( (5,4) \) \( W=\square \)

Compute the divergence of \( F(x, y)=1.2 x i+1.75 y j \) \( \operatorname{div} F=\square \)

52. \( \star \sum_{n=1}^{\infty} \sqrt{n}(\sin n)\left(\sin \frac{1}{n}\right)^{2} \)

Evaluate the line integral. \( \int_{C} 5 z d s \), where \( C \) is the line segment from \( (5,0,1) \) to \( (6,-6,6) \) \( \int_{C} 5 z d s=\square \)