Integrating: Solving the Problem of Ln(u) in Answer

In summary, to solve the integral \int(\frac{x}{\sqrt{1-x^{2}}})dx, you can make the substitution u = 1 - x^2 and use the formula \int(\frac{1}{u})du = ln(u) + C to get the final answer of -ln(\sqrt{1-x^2}) + C. Another substitution that works is u = sqrt(1-x^2), which leads to the final answer of -sqrt(1-x^2) + C.
  • #1
28
0

Homework Statement



[tex]\int(\frac{x}{\sqrt{1-x^{2}}})dx[/tex]

Homework Equations





The Attempt at a Solution



My calculator tells me that the answer should be -sqrt(1-x^2) but if I pick u = sqrt(1-x^2), then dx = (sqrt(1-x^2)*du)/x, which leaves me with -integral((sqrt(1-x^2)/u)du), the problem I am having is getting rid of the "ln(u)" in my final answer, any suggestions?
 
Physics news on Phys.org
  • #2
If u^2=1-x^2
2u du =-2x dx => - u du = x dx

Now you'd just get

[tex]\frac{-u}{u} du[/tex]
 
  • #3
ok got it, thank you
 
  • #4
rock.freak667 said:
If u^2=1-x^2
2u du =-2x dx => - u du = x dx

Now you'd just get

[tex]\frac{-u}{u} du[/tex]
Another substitution that works is u = 1 - x^2, du = -2xdx.
The integrand then becomes -(1/2)du/u^(1/2), which is also an easy one to integrate.
 

1. What is the problem with solving for Ln(u)?

The main problem with solving for Ln(u) is that it is an inverse function, meaning that it is not easily solved using traditional algebraic methods. This is because the natural logarithm function is defined as the inverse of the exponential function, and therefore requires a different approach to solve for u.

2. How can I solve for Ln(u) in my calculations?

To solve for Ln(u), you can use the properties of logarithms to simplify the expression. One common approach is to use the fact that Ln(u) is equal to the exponent in the exponential function e^x, so you can rewrite the equation as u = e^Ln(u). This allows you to use the rules of exponents to simplify the expression and solve for u.

3. Can I use a calculator to solve for Ln(u)?

Yes, most calculators have a natural logarithm function (usually denoted as "ln" or "log") that you can use to solve for Ln(u). Simply input the expression and the calculator will give you the solution.

4. Are there any special cases when solving for Ln(u)?

Yes, there are a few special cases to keep in mind when solving for Ln(u). One is when u is equal to 1, in which case Ln(u) is equal to 0. Another is when u is negative, in which case the natural logarithm is undefined. Lastly, when u is equal to 0, the natural logarithm is undefined as well.

5. Can I solve for Ln(u) using calculus?

Yes, you can use calculus to solve for Ln(u). One approach is to use the derivative of the natural logarithm function (which is 1/u) to find the inverse function of Ln(u). Another approach is to use the integral of 1/u to find the inverse function of Ln(u).

Suggested for: Integrating: Solving the Problem of Ln(u) in Answer

Back
Top