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How large should n be to guarantee the Trapezoidal Rule/Midpoint Rule/Simpson's Rule approximation for the integral 0 to pi/2 of sqr(cosx) dx be accurate to within 0.000001? To find n, I know I need to use the error bound formula for the trapezoidal, midpoint, or simpson's rule. I found the...

I have changed K(k)=(integral 0 to pi/2) 1/sqrt(1-k^2sin^2x) dx to K(k)=(integral 0 to 1) 1/[2cosx(sqrt(t-k^2t^2))] dt and E(k)=(integral 0 to pi/2) sqrt(1-k^2sin^2x) dx to E(k)=(integral 0 to 1) [sqrt(1-k^2t)]/[2(sqrt(t))cosx] dt by performing the change t=sin^2(x), dt=2sinxcosx dx and...

How can I start to prove the equation: K'(k) = E(k)/k(1 − k^2)−K(k)/k for all 0 < k < 1 by using the complete elliptic integrals of the first and second kind? I have already changed the integral from (0 to pi/2) to (0 to 1) for both K(k) and E(k), but I'm not sure where to go from there. Thank you!

Proving, or providing a counterexample: "If f is a continuous function on [0, ∞) and f is unbounded, then the improper integral 0 to ∞ f(x)dx has to diverge." I know that this statement is false, but how can I go about providing a counterexample. I know that f(x) would have to be piecewise in...