4. (10 marks). Suppose f is a differentiable and one-to-one function. (a) Use integration by parts...

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4. (10 marks). Suppose f is a differentiable and one-to-one function. (a) Use integration by parts to prove that - dx [ ^'(x) dx = x(x) [ (1(x)) dr (b) With y=f-1(x), show that f-1(x) - f(y) dy [(x) dx = x f(x) [f(y) (c) Use part (b) to evaluate Ja arctan(r)dr 4. (10 marks). Suppose f is a differentiable and one-to-one function. (a) Use integration by parts to prove that - dx [ ^'(x) dx = x(x) [ (1(x)) dr (b) With y=f-1(x), show that f-1(x) - f(y) dy [(x) dx = x f(x) [f(y) (c) Use part (b) to evaluate Ja arctan(r)dr