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Prove by induction that the sum of the first n integers can be written as (1/2)(n)(n+1).

For n = 1, the sum is given by (1/2)(1)(1+1), which gives 1, the expected result. We now assume that the statement is true for some k. If we look at k+1, the sum is given by 1 + 2 + ... + k + (k+1). Since we...
JS
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Integral of ln x

xln(x) - x
OG
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y = artanh(x/sqrt(1+x^2)) , find dy/dx

Some of these examiners quite like asking students to find the derivative of an inverse trig or hyperbolic function to try and catch someone off guard. The best way to approach these is to first multiply bot...
EL
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Show that the sum from 1 to n of 1/(2n+1)(2n-1) is equal to n/(2n+1) by Induction

First we check that this is true for n=1: S 1 = 1/(1x3) which is equal to n/(2n+1) for n=1 therefore S n = n/(2n+1) is true for n = 1. Next assume that it is true for n=k. S k = k/(2k+1). Now using this assu...
JF
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Two planes have eqns r.(3i – 4j + 2k) = 5 and r = λ (2i + j + 5k) + μ(i – j – 2k), where λ and μ are scalar parameters. Find the acute angle between the planes, giving your answer to the nearest degree.

Summary of solution: To find the angle between the planes, we must find the normal vector to each plane and then use the scalar product to find the angle between these two normal vectors. Breaking this down ...
DC
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