mymaddy
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Calculus Question - Integration
Integrate sqrt((1-x)/x)dx
I think I tried everything and I have spent more than an hour now. :(
Please help.
I think I tried everything and I have spent more than an hour now. :(
Please help.
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Wait, I'm partially confused. :)
(1-x)/x is positive only when x is in (0, 1), so the sqrt is real only in the same range. Still, sqrt(x-1) is imaginary in this range.
(1-x)/x is positive only when x is in (0, 1), so the sqrt is real only in the same range. Still, sqrt(x-1) is imaginary in this range.
sqrt((1-x)/x) (x-log(sqrt(x-1)+sqrt(x))s qrt(x/(x-1 ))
= x sqrt((1-x)/x) - i log(sqrt(x-1)+sqrt(x))
^
hmm, indeed
= x sqrt((1-x)/x) - i log(sqrt(x-1)+sqrt(x))
^
hmm, indeed
log(ai+b) = log(sqrt(a^2+b^2)) + i*atan2(a,b)
Assuming 0 < x < 1
sqrt(x-1) = i*sqrt(1-x)
a=sqrt(1-x), b = sqrt(x)
log(sqrt(1-x)*i + sqrt(x)) = log(sqrt(1-x + x)) + i*theta
= log(sqrt(1)) + i*theta
= i*theta,
So it looks like the log term is purely imaginary when 0 < x < 1, so multiplying it by i makes it real.
given that 0 < x < 1,
theta = asin(sqrt(1-x) / sqrt(sqrt(1-x)^2 + sqrt(x)^2))
= asin(sqrt(1-x) / sqrt(1 - x + x))
= asin(sqrt(1-x))
So, it looks like we can replace the i*log with asin(sqrt(1-x))
integral = x*sqrt((1-x)/x) - asin(sqrt(1-x))
Let's take the derivitive and see what we get.
d{sqrt((1-x)/x)) = d{sqrt(1-x) / sqrt(x) } =
(sqrt(x)*(-0.5/sqrt(1-x)) - sqrt(1-x)*(0.5/sqrt(x))) / x =
multiply num and denom by sqrt(x)*sqrt(1-x)
(x*(-0.5) - (1-x)*(0.5))/(x*sqrt(x)*sq rt(1-x)) =
(-x/2 - 1/2 + x/2) =
1/(2*x*sqrt(x)*sqrt(1-x))
d{x*sqrt((1-x)/x)} = 1/(2*sqrt(x)*sqrt(1-x)) + sqrt((1-x)/x)
d{asin(f(x))} = f'(x) * 1/sqrt(1 - f(x)^2)
d{asin(sqrt(1-x))} = (-0.5/sqrt(1-x)) * 1/sqrt(1 - (1-x))
= (-0.5/sqrt(1-x)) / sqrt(x)
= - 1/(2*sqrt(x)*sqrt(1-x))
adding it all together:
d{x*sqrt((1-x)/x) - asin(sqrt(1-x))}
= 1/(2*sqrt(x)*sqrt(1-x)) + sqrt((1-x)/x) - 1/(2*sqrt(x)*sqrt(1-x))
= sqrt(1-x)/x
So I believe that shows that the integral of sqrt(1-x)/x is x*sqrt((1-x)/x) - asin(sqrt(1-x)) + C
Assuming 0 < x < 1
sqrt(x-1) = i*sqrt(1-x)
a=sqrt(1-x), b = sqrt(x)
log(sqrt(1-x)*i + sqrt(x)) = log(sqrt(1-x + x)) + i*theta
= log(sqrt(1)) + i*theta
= i*theta,
So it looks like the log term is purely imaginary when 0 < x < 1, so multiplying it by i makes it real.
given that 0 < x < 1,
theta = asin(sqrt(1-x) / sqrt(sqrt(1-x)^2 + sqrt(x)^2))
= asin(sqrt(1-x) / sqrt(1 - x + x))
= asin(sqrt(1-x))
So, it looks like we can replace the i*log with asin(sqrt(1-x))
integral = x*sqrt((1-x)/x) - asin(sqrt(1-x))
Let's take the derivitive and see what we get.
d{sqrt((1-x)/x)) = d{sqrt(1-x) / sqrt(x) } =
(sqrt(x)*(-0.5/sqrt(1-x)) - sqrt(1-x)*(0.5/sqrt(x))) / x =
multiply num and denom by sqrt(x)*sqrt(1-x)
(x*(-0.5) - (1-x)*(0.5))/(x*sqrt(x)*sq
(-x/2 - 1/2 + x/2) =
1/(2*x*sqrt(x)*sqrt(1-x))
d{x*sqrt((1-x)/x)} = 1/(2*sqrt(x)*sqrt(1-x)) + sqrt((1-x)/x)
d{asin(f(x))} = f'(x) * 1/sqrt(1 - f(x)^2)
d{asin(sqrt(1-x))} = (-0.5/sqrt(1-x)) * 1/sqrt(1 - (1-x))
= (-0.5/sqrt(1-x)) / sqrt(x)
= - 1/(2*sqrt(x)*sqrt(1-x))
adding it all together:
d{x*sqrt((1-x)/x) - asin(sqrt(1-x))}
= 1/(2*sqrt(x)*sqrt(1-x)) + sqrt((1-x)/x) - 1/(2*sqrt(x)*sqrt(1-x))
= sqrt(1-x)/x
So I believe that shows that the integral of sqrt(1-x)/x is x*sqrt((1-x)/x) - asin(sqrt(1-x)) + C
hello
it is very easy: look!
integral : ( x^2 - 2*x +1 ) / (x^2) dx
now divide ( x^2 - 2*x +1 ) to (x^2) . so the question will have another form:
integral : [ 1 - 2/x + 1/(x^2) ] dx
well, now we can easily solve it:
equals with : x - 2*Ln(x) - a/x + Constant
or
equals with : x - Ln(x^2) - a/x + Constant
the both are correct, no difference. Regards!
it is very easy: look!
integral : ( x^2 - 2*x +1 ) / (x^2) dx
now divide ( x^2 - 2*x +1 ) to (x^2) . so the question will have another form:
integral : [ 1 - 2/x + 1/(x^2) ] dx
well, now we can easily solve it:
equals with : x - 2*Ln(x) - a/x + Constant
or
equals with : x - Ln(x^2) - a/x + Constant
the both are correct, no difference. Regards!
hello
it is very easy: look!
integral : ( x^2 - 2*x +1 ) / (x^2) dx
now divide ( x^2 - 2*x +1 ) to (x^2) . so the question will have another form:
integral : [ 1 - 2/x + 1/(x^2) ] dx
well, now we can easily solve it:
equals with : x - 2*Ln(x) - a/x + Constant
or
equals with : x - Ln(x^2) - a/x + Constant
the both are correct, no difference. Regards!
it is very easy: look!
integral : ( x^2 - 2*x +1 ) / (x^2) dx
now divide ( x^2 - 2*x +1 ) to (x^2) . so the question will have another form:
integral : [ 1 - 2/x + 1/(x^2) ] dx
well, now we can easily solve it:
equals with : x - 2*Ln(x) - a/x + Constant
or
equals with : x - Ln(x^2) - a/x + Constant
the both are correct, no difference. Regards!
What are you on, CS?
>> integral : ( x^2 - 2*x +1 ) / (x^2) dx
This is not what we're trying to integrate;
this is:
>>Integrate sqrt((1-x)/x)dx
>> integral : ( x^2 - 2*x +1 ) / (x^2) dx
This is not what we're trying to integrate;
this is:
>>Integrate sqrt((1-x)/x)dx
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OOPS... ? should have been ² (square)....
ASKER
Thanks again!