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How do you solve hard integration questions using information you know

lets try a few integrations. what is integral of dx/(1+x)? You can see that it is in the form of f'(x)/f(x) so it is ln[f(x)] right? now what about integral of (e^x/(1+e^x))dx that is also in the form of f'(...
DA
Answered by Desmond A. • MAT tutor
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Prove by induction that (n^3)-n is divisible by 3 for all integers n>0 (typical fp1 problem)

Let P(n) be the statement “(n^3)-n is divisible by 3” First we’ll examine the base case: P(1) (n^3)-n=1^3-1=1-1=0 0=3*0, so is divisible by 3, and so P(1) is true Now assume for some k>0, an integer, that...
TN
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A football is kicked at 30 m/s at an angle of 20° to the horizontal. It travels towards the goal which is 25 m away. The crossbar of the goal is 2.44 m tall. (A) Does the ball go into the goal, hit the crossbar exactly, or go over the top?

With questions about projectiles, such as a ball travelling in the shape of a parabola through the air as in the above question, it is always good to split the problem into the horizontal and vertical compon...
ES
Answered by Ehsaan S. • Maths tutor
14643 Views

Explore Shakespeare’s presentation of family in King Lear. You must relate your discussion to relevant contextual factors and ideas from your critical reading.

Shakespeare illustrates that at the beginning of the play, Lear’s views on how family love is expressed and measured is corrupt as his two insincere daughters are rewarded while his most loving and most love...
BS
Answered by Bethan S. • English tutor
5561 Views

Consider f:R -> R, f = x/ sqrt(x^2+1). Prove that for any a between -1 and 1, f(x)=a has only one solution.

f'(x)=( sqrt (x^2+1) - x * ( x / sqrt (x^2 +1) ) ) / (x^2+1) = (x^2 + 1 + x^2) / ( (x^2 + 1) * sqrt ( x^2 + 1) ) = 1 / ( (x^2 + 1) * sqrt (x^2 + 1) ). f'(x) > 0 for any x => f is increasing. When x-&gt...
AC
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