Thread: Integration, Proof and limit.

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I've found the integral, but how do i prove?

thank you.

Of the three terms in the inequality, which one corresponds to the exact value of the integral?

middle one.

Right. Is the integrand increasing or decreasing as x increases?

Draw the graph of y= 1/x between x= k and x= k+1. Look at the area of the rectangle with base from k to k+1 and height 1/k, the area of the rectangle with base from k to k+ 1 and height 1/(k+1), and the area under the graph.

By the mean value theorem, there exists a D in [K,K+1] such that

k_<D<_k+1 so 1/(k+1)_<1/D<_1/k so the required result follows.

Originally Posted by BabyMilo

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The following is found in most calculus textbooks in a list of properties of integrals.

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Now notice that