Integration?

Question

The integration of X'/X is ln IXI + c.

Then what is the integration of X/X' ?

Can anyone help pls? Thanks!

Answer 1

There is no general formula for the integral of X/X'.

Answer 2

I think you've mixed this up a bit. The first line seems to me to suggest the rule
integral of f'(X)/f(X) is ln IXI + C

For example, if you want to integrate
(6x - 5)/(3x^2 - 5x + 4), notice that if the denominator is f(x), then the numerator is f'(x), so the integral is
ln (3x^2 - 5x + 4) + C [I didn't bother to put absolute value because f(x) is always positive]

However if you were trying to integrate
(x^2 - 3)/(x^3 - 9x + 12), the derivative of the bottom is
3x^2 - 9 which is not the numerator. However, in this case it's just 3 times the numerator, so we write the expression as
(1/3)*(3x^2 - 9)/(x^3 - 9x + 12) and the integral is

(1/3)*ln I(x^3 - 9x + 12)I

There isn't any rule that I know of for integrating f(x)/f'(x).

In the first example I gave, that would be
(3x^2 - 5x + 4)/(6x - 5).

If you know how to divide a polynomial, you find this is equal to
x/2 - 5/12 - 1/(36x - 30) and its integral is

(x^2)/4 - 5x/12 - (1/36)*ln I(36x-30)I
No special relevance of the bottom being the derivative of the top.

Answer 3

integration is a form of calculus discovered over 4000 years ago in Greece by Euclid. The original use of intergration was an easier method of finding the area enclosed under a curve.

For example, for the curve X'/X dX (meaning with respect to the X varible) the evaluation of the integeral results in the ln(|X|) + a constant C.

For the example you asked for X/X' dX the integration of the curve is best understood if you were to rewrite the equation as:

X^n/(X^n)' dX where n is the power of the term X

the integration of the equation is then

X^(n+1)/(X^n x (n+1)) if and only if X^n' is a constant value (meaning that X is raised to the power of 1 and X' is to the power of 0)

If this is not the case the evaluation of the integral is much more complex and may require some fancy foot work, or a decent table of the integrals.

Integral calculus is something you may be exposed to in highschool, or in a sophmore year in college , depending on your education path.

I really hope I have made this at least as clear as mud for you. Please feel free to contact me if you have a more specific problem.

Ken

Answer 4

if X` is a constant then:
Integration of X is X^2 / 2
so Integration of X/X` = X^2 / 2X`

Answer 5

X' in general is obviously not a constant.