Chapter 16: Integrals and Vector Fields - Practice Exercises - Page 1029: 34

$$ e^y+\sin (xz)+C $$

A vector field is said to be conservative when $ curl F=\nabla \times F=0$ Now $$ f(x,y,z)=\int_0^x f_1 (t,0,0) dt+ \int_0^y f_2 (x,t,0) dt+\int_0^z f_3 (x,y,t) dt \\=\int_0^x (0) \times \cos (t \cdot 0) dt+ \int_0^y (e^t) dt+\int_0^z x \cos x \space t \space dt\\=[e^t]_0^y+x(\dfrac{1}{x}) \times [\sin t]_0^x+C \\=e^y+\sin (x \space z)+C $$