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The general form of the partial fractions we want is A/(x-1) + B/(2x-3), where A and B are constants we need to find the values of.
We know that A/(x-1) + B/(2x-3) = (2x-1)/(x-1)(2x-3), so we can conclude that A(2x-3) + B(x-1) = 2x - 1
The best way to solve for A and B now is to equate the coefficients on each side of the equation. On the left, we have 2A + B x's, while we have 2x on the right. This means that 2A + B = 2.
Similarly, we have that -3A - B = -1.
We now have a pair of simultaneous equations to solve in A and B. If we add together the above equations, we get -A = 1, which implies that A = -1. We can then obtain that B = 2. Putting these values into the general equation for the partial fractions gives -1/(x-1) + 2/(2x-3), which is our answer.see more
The key here is to realise that tanx = sinx/cosx. If we write out the left hand side of the equation in terms of sine and cosine we get:
cosx/sinx + sinx/cosx
These two fractions can be put over a common denominator of sinxcosx to give:
(cos2x + sin2x)/sinxcosx
If we then use the well-known identity cos2x + sin2x = 1, we see that the above expression is equivalent to 1/sinxcosx, which is the expression we were required to find.see more
Since there is an x written as a power here, it suggests that this question should be solved using logs (here assume I am using the natural logarithm, which has base e). Taking logs of both sides of the equation gives log5x = log8
Using the rule logab = bloga, the equation becomes xlog5 = log8
If we then divide the equation through by log5, we see that x = log8/log5 = 1.29see more