Answer
$$20\pi $$
Work Step by Step
$$\eqalign{
& \int_0^{2\pi } {\left( {10 + 10\cos x} \right)} dx \cr
& {\text{sum rule for integrals}} \cr
& = 10\int_0^{2\pi } {dx} + 10\int_0^{2\pi } {\cos x} dx \cr
& {\text{integrate by using the basic integration rule }}\int {\cos x} dx = \sin x + C\,\,\,\left( {{\text{page 692}}} \right) \cr
& = 10\left( {10x} \right)_0^{2\pi } + 10\left( {\sin x} \right)_0^{2\pi } \cr
& {\text{use fundamental theorem of calculus }}\int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\,\,\,\left( {{\text{see page 388}}} \right) \cr
& = 10\left( {2\pi - 0} \right) + 10\left( {\sin \left( {2\pi } \right) - \sin \left( 0 \right)} \right) \cr
& {\text{simplifying}} \cr
& = 20\pi + 10\left( {0 - 0} \right) \cr
& = 20\pi \cr} $$
