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Mermaid

Math

Let f ⁣:[a,b]Rf\colon[a,b]\to\R be Riemann integrable. Let F ⁣:[a,b]RF\colon[a,b]\to\R be F(x)=axf(t)dtF(x)=\int_{a}^{x} f(t)\,dt. Then FF is continuous, and at all xx such that ff is continuous at xx, FF is differentiable at xx with F(x)=f(x)F'(x)=f(x).

I=02πsin(x)dxI = \int_0^{2\pi} \sin(x)\,dx