Consider the integral ( x ,dx

MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the integral ∫arcsin⁡x dx.\displaystyle \int \arcsin x\,dx. Work through the following tasks, providing complete justification for each step. (1) Find an explicit formula for ∫arcsin⁡x dx\displaystyle \int \arcsin x\,dx expressed solely in terms of elementary functions (no integral signs should remain). (2) Differentiate your result from part (a) to verify that it indeed yields arcsin⁡x\arcsin x as its derivative. (3) Using your antiderivative, determine the exact value of ∫01/2arcsin⁡x dx.\displaystyle \int_{0}^{1/2} \arcsin x\,dx. Leave any inverse trigonometric constants unevaluated (for example, keep arcsin⁡(1/2)\arcsin(1/2) rather than converting it to π/6\pi/6 only if it is convenient).

Related theory

Browse the problem library