what method of integration is this?
I think it’s usually called ‘integration by partial fractions’ or something similar. Here’s a pretty good tutorial: https://www.intmath.com/methods-integration/11-integration-partial-fractions.php
Hi, thanks for the comments. You are absolutely right, both answers have silly mistakes in them.
In part (i) the answer correctly derives the simultaneous equations a + b = 1 and -a + b = 2, but the solution to these should be b = 3/2 and a = -1/2, and this makes your final answer correct.
In part (ii) there is an attempt to use the substitution u = x + 1, but the substitution doesn’t actually get made. If you do the substitution properly, the integral becomes Integral from 1 to 2 of (1 / sqrt(x)) dx. This gives us 2 sqrt(2) – 2 sqrt(1), which is 2(sqrt(2) – 1), as you say in your comment.
Thanks very much for your corrections; it’s so easy for careless errors to creep in to these answers and it’s really useful to get contributions from people who spot them.
could the first answer be represented as 1/2(In|(x-1)^3/(x+1)|)+c?
Hope you can clarify this for me please. And if its correct is it fine to leave the c outside because physicsmathstutor has this solution which confused me a little: http://www.physicsandmathstutor.com/pat/solutions-2011/
Yes it is fine to leave the c outside. Remember that ln(x) + ln(y) = ln(xy), so ln(x) + c = ln(Ax) where ln(A) = c. So you could rewrite my answer as ln(A(x-1)^3/(x+1))/2. This is the same form as the answer you refer to in the link.
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