Apparently my result is correct. Nevertheless, can someone take a look at the final expression and judge if the absolute values inside logarithm have been correctly reduced?
Can anybody spot the mistake? I have no idea where it is... And it is very trivial example...
$\gamma = \int \frac{\sqrt{x}+4}{x\sqrt{x} - x}dx = \Bigg(\Bigg( t = \sqrt{x}, t^2=x, 2tdt=dx \Bigg)\Bigg) = 2\int\frac{(t+4)tdt}{t^3-t^2} = 2\int\frac{t+4}{t^2-t}dt = \int\frac{2t+8}{t(t-1)}dt = \gamma$
Partial fraction decomposition:
$\Rightarrow \frac{2t+8}{t(t-1)} = \frac{A}{t} + \frac{B}{t-1}$
$2t + 8 = A(t-1) + Bt$
$2t+8=At - A + Bt$
$2 = A + B \quad \land \quad 8 = -A$
$\Longrightarrow A = -8 \quad \land \quad B = 10$
$\gamma = -8\int\frac{dt}{t} + 10\frac{dt}{t-1} = -8\ln{|t| + 10\ln{|t-1|} + C}$
After substitution:
$\gamma = -8\ln|\sqrt{x}| + 10\ln{|\sqrt{x} - 1|} = \ln{\frac{(\sqrt{x}-1)^{10}}{x^4}} + C$
Two questions:
where's the mistake? I cannot seem to find it, wolframalpha says the result is wrong
let's assume that the result is good (it is not, but let's assume). Can someone look at the very final result and tell me if the absolute values inside natural logarithm were reduced in a good way, or I missed some absolute values in there?
Thanks
WolframAlpha does, in fact, agree with your answer.
As far as considering the absolute values for your answer: with your simplification, your result remains equivalent. In particular, we have $$|\sqrt{x}-1|^{10}=(\sqrt{x}-1)^{10},\qquad |x|^4=x^4.$$ This is because the base has been raised to an even exponent; for any $y\in\Bbb R$, we have $y^{2k}\geq0$ for $k\in\Bbb N$.