SOLVED:Given a continuous function f on an interval I, prove-using ordinary integration of functions - that there exists a unit-speed curve β(s) in 𝐑^2 for which f(s) is the plane curvature. (Hint: Reverse the logic in Ex. 8 of Sec. 2.3.)

What does it mean for a function to be continuous on an interval? | Study Prep in Pearson

SOLVED: 56. a) Let f(x) be a continuous function defined on the interval [0, a], where a is a positive constant. Prove that ∫f(x)dx = ∫(a-x)dx / (1+sinx + cosx) (b) (i) Prove that ∫sinx / (1+cosx) dx = 0 + ∫sinx (ii) Prove that ∫1+cosx / (1+sinx) dx (c) -Find ∫dx / (1+sinx) xsinx + cosx + 1 dx / (1+cosx)(1+sinx) (d) Using integration by parts, evaluate ∫xsinx+cosx+1 dx