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ehrenfest
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Homework Statement
What technique should I use to evaluate:
[tex] \int_0^L \frac{1}{(x^2+z^2)^{3/2}} dx [/tex]
Is there a u-substitution that works?
Homework Equations
The Attempt at a Solution
Last edited:
bob1182006 said:With respect to what variable are you integrating? x or z?
bob1182006 said:Try pulling out the variable you're integrating w.r.t. and you should be able to do a u-substitution then.
bob1182006 said:Pull out x from the ()^3/2 term, and do a you can do a u-substitution from there.
bob1182006 said:you changed the limits, your original integral is from 0 to L, so the new limits should be from 0 to 1+(z^2)/L^2
Do you really mean hyperbolic sine?Gib Z said:[itex]x= z \sinh u[/itex] works out fastest here, I can do it in my head with that one.
bob1182006 said:[tex]\frac{1}{(x^2+z^2)^{3/2}}=\frac{x^{-3}}{(1+z^2x^{-2})^{3/2}}[/tex]
u-sub the 1+z^2... and you'll get a 1/u^3/2 integral
You get the same answer as with a trig substitution, but you will have to multiply by x/x in order to match the answer from trig substitution.
Gib Z said:[itex]x= z \sinh u[/itex] works out fastest here, I can do it in my head with that one.
A u-substitution is a method used in calculus to simplify integrals by changing the variable of integration. It involves substituting a new variable, usually denoted as u, in place of the existing variable in the integral. This allows for the integral to be rewritten in a simpler form, making it easier to solve.
A u-substitution is most commonly used when the integrand (the expression inside the integral) contains a function and its derivative, such as f(x) and f'(x). It can also be used when there is a product of functions, a chain rule, or a trigonometric function present in the integrand.
The general steps for performing a u-substitution are as follows:
One common mistake when using u-substitution is forgetting to replace the differential (dx) with the appropriate differential (du) after substituting in the new variable. It is also important to be careful with the signs when substituting and integrating, as this can often lead to incorrect results.
Yes, there is a shortcut for performing u-substitution called the "u-substitution formula." This formula states that if the integrand contains a function f(x) and its derivative f'(x), then the integral can be rewritten as ∫f(x)dx = ∫f(u)du. This can save time and effort when performing u-substitution, but it is important to still follow the steps and avoid common mistakes.