Source: A question bank of "tough" problems on integrals (maybe tough for a noob like me). Started by learning integration for use in physics only, but now it's got me hooked :p
Problem: Evaluate the indefinite integral $$\int {x\,\mathrm{d}x\over(7x-10-x^2)^{3/2}}.$$
I have used all the tools in my arsenal; substitution: no viable substitution comes in mind here. I have tried factoring the quadratic but that doesn't help.
I have tried to multiply-divide the denominator by $x^2$ and then substitute $x={1\over t}$ but no help. I'm actually stuck right now. Please give me a hint to solve this one. All help appreciated!
@Frank gave it a shot as well...
$$\int {x\,\mathrm{d}x\over{(-1)^{3/2}(x^2-7x+10)^{3/2}}}.$$
$$\int {x\,\mathrm{d}x\over{(i)^3(x^2-7x+10)^{3/2}}}.$$ ($i$ is the imaginary unit)
Clearly we don't get any imaginary term in the answer and there are probably no chances that we'll cancel the imaginary number. That's why I did not look forward to this method. Will go ahead and try the Euler substitution...
Edit: This question is solved but I'm still looking for a better, more faster alternative as Euler's substitution can sometimes invite a bunch of calculations.
Answer
Thanks @DrSonnhardGraubner for giving me the right article for the problem. I didn't know about this one.
We are going to use the third substitution of Euler here, wherein we assume that
$$\sqrt{7x-10-x^2} = (5-x)t$$ (consider factorization)
$$t = \sqrt{(x-2)\over(5-x)}$$
partially differentiate to get an expression of $\mathrm{d}x$ in terms of $\mathrm{d}t$
$$\mathrm{d}x = \frac{6t \mathrm{d}t}{(t^2+1)^2}$$
Now substitute $x$ with a function of $t$ according to the above equation and get the answer as given above in the comment by @John Chessant
$$-\frac{2}{9}\cdot\frac{20-7x}{\sqrt{(2-x)(5-x)}}+C$$
Any alternates are welcome!
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