Answer
$$ - 11\left( {x\sin x + \cos x} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\left( { - 11x} \right)\cos x} dx \cr
& {\text{setting }}\,\,\,\,\,\,u = - 11x{\text{ then }}du = - 11dx\,\,\,\,\,\,\,\,\,\, \cr
& {\text{and}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,dv = \cos xdx{\text{ then }}v = \sin x \cr
& {\text{Substituting these values into the formula for integration by parts}} \cr
& \int u dv = uv - \int {vdu} \cr
& \int {\left( { - 11x} \right)\cos x} dx = \left( { - 11x} \right)\left( {\sin x} \right) - \int {\left( {\sin x} \right)\left( { - 11dx} \right)} \cr
& {\text{simplifying}} \cr
& = - 11x\sin x + 11\int {\sin x} dx \cr
& {\text{integrate by using the Basic Trigonometric integral }}\int {\sin x} dx = - \cos x + C \cr
& = - 11x\sin x - 11\cos x + C \cr
& {\text{factoring}} \cr
& = - 11\left( {x\sin x + \cos x} \right) + C \cr} $$
