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Differention

Does ∫(x^-2 - 1)^-0.5 dx =

1. 2(x^-2 - 1)^0.5 + C

2. -x^3(x^-2 - 1)^0.5 + C

3. something else, if so, what
Reply 1
Avatar for HTale
HTale
14
dhokes
Does ∫(x^-2 - 1)^-0.5 dx =

1. 2(x^-2 - 1)^0.5 + C

2. -x^3(x^-2 - 1)^0.5 + C

3. something else, if so, what


Do you mean
∫1(1/x2)−1\displaystyle \int \frac {1}{\sqrt (1/x^2) - 1} ?
Reply 2
Avatar for Gaz031
Gaz031
15
∫11x2−1dx=∫x1−x2dx\int \frac{1}{\sqrt{\frac{1}{x^2}-1}}dx=\int \frac{x}{\sqrt{1-x^2}} dx (multiplying the numerator and denominator of the main fraction by x)
You can then perform the substitution u=1−x2u=1-x^2 to perform the integration where its then merely a matter of putting it in a similar form to (1), (2) (with a negative power of x inside the bracket) and assessing whether it qualifies for (1), (2) or (3).
Reply 3
Avatar for dhokes
dhokes
OP
13
HTale
Do you mean
∫1(1/x2)−1\displaystyle \int \frac {1}{\sqrt (1/x^2) - 1} ?


yes, and its the sqrt of (1/x^2 - 1)
Reply 4
Avatar for HTale
HTale
14
Gaz031
∫11x2−1dx=∫x1−x2dx\int \frac{1}{\sqrt{\frac{1}{x^2}-1}}dx=\int \frac{x}{\sqrt{1-x^2}} dx (multiplying the numerator and denominator of the main fraction by x)
You can then perform the substitution u=1−x2u=1-x^2 to perform the integration where its then merely a matter of putting it in a similar form to (1), (2) (with a negative power of x inside the bracket) and assessing whether it qualifies for (1), (2) or (3).


Id say a trigonometric substitution is in order :smile:
Reply 5
Avatar for yazan_l
yazan_l
5
it's:
Unparseable latex formula:

\int (x^{-2} - 1 ) ^{-\frac{1}{2}} dx = \\[br]\int \frac {1}{\sqrt{x^{-2}-1}} dx = \\[br]\int \frac {1} {\sqrt{\frac{1}{x^{2}}-1}} dx =\\[br]\int \frac {1}{\sqrt{\frac{1-x^{2}}{x^{2}}}} dx = \\[br]\int \frac {x}{\sqrt{1-x^{2}}} dx \\[br]\\[br]u = 1-x^{2}\\[br]\frac {du}{dx} = -2x\\[br]dx = -\frac{1}{2x} du\\[br][br]\int \frac {x}{\sqrt{1-x^{2}}} dx =\\[br]\int \frac {x}{\sqrt{u}} . -\frac{1}{2x} du = \\[br]-\frac{1}{2} \int u^{-\frac{1}{2}} du =\\[br]- ( \sqrt{1-x^{2}} ) + C

Reply 6
Avatar for yazan_l
yazan_l
5
why did this think take my post to the last place when i edited it??? when i posted it it was just after Gaz's post!!!
Reply 7
Avatar for yazan_l
yazan_l
5
and by the way: this is "Integration" not "Differentiation" and above all not "Differention" :biggrin: :tsr:
cheers
Reply 8
Avatar for HTale
HTale
14
Here's a solution using a different substitution
Solution.pdf21.9KB
Reply 9
Avatar for dhokes
dhokes
OP
13
yazan_l
and by the way: this is "Integration" not "Differentiation" and above all not "Differention" :biggrin: :tsr:
cheers


lol big mistake, btw HTale, what program do u use to create pdfs??

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