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Suppose f and 9 aC diflerentiable OH [a.6] and that f and 9 are Riemann integrable on this interval. Prove thatf(r)g' (r)dr = [f(6)g(b) - f(a)g(a)]f' (r)g...

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Suppose f and 9 aC diflerentiable OH [a.6] and that f and 9 are Riemann integrable on this interval. Prove thatf(r)g' (r)dr = [f(6)g(b) - f(a)g(a)]f' (r)g(r)dz.Hint: Use the Fundamental Theorem of Calculus and the fact that (fg)' f"g + fg'

Suppose f and 9 aC diflerentiable OH [a.6] and that f and 9 are Riemann integrable on this interval. Prove that f(r)g' (r)dr = [f(6)g(b) - f(a)g(a)] f' (r)g(r)dz. Hint: Use the Fundamental Theorem of Calculus and the fact that (fg)' f"g + fg'



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Prove that the function $f$ defined by $$f(x)=\left\{\begin{array}{ll}1 & \text { if } x \text { is rational } \\0 & \text { if } x \text { is irrational }\end{array}\right.$$.is not integrable on [0,1] . Hint: Show that no matter how small the norm of the partition, $\|P\|$, the Riemann sum can be made to have value either 0 or 1.


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