Consider the integral ∫arcsinxdx. Work through the following tasks, providing complete justification for each step.
a. Find an explicit formula for ∫arcsinxdx expressed solely in terms of elementary functions (no integral signs should remain).
b. Differentiate your result from part (a) to verify that it indeed yields arcsinx as its derivative.
c. Using your antiderivative, determine the exact value of ∫01/2arcsinxdx. Leave any inverse trigonometric constants unevaluated (for example, keep arcsin(1/2) rather than converting it to π/6 only if it is convenient).