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walksintoabar
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Homework Statement
integral of 1/(x^2 + z^2)^(3/2) dx,
where z is a constant
Homework Equations
The Attempt at a Solution
I set u = arctan(x/z) so du = z/(x^2 + z^2) dx but now I'm honestly stuck.
Using u substitution with arctan in integration involves rewriting the integral in terms of a new variable u, then finding the derivative of u with respect to x. This derivative is substituted in place of the dx in the original integral, and the resulting integral is evaluated using the properties of arctan.
U substitution with arctan is useful in integration because it allows us to simplify complex integrals involving trigonometric functions. By substituting in a new variable, we can often transform the integral into a simpler form that can be evaluated more easily.
The key steps to follow when using u substitution with arctan are: identify the appropriate substitution, find the derivative of u with respect to x, substitute in the derivative and rewrite the integral in terms of u, evaluate the integral using the properties of arctan, and finally, replace u with the original variable to obtain the final solution.
Some common mistakes to avoid when using u substitution with arctan include forgetting to find the derivative of u with respect to x, not substituting the derivative in the integral, and forgetting to replace u with the original variable in the final solution. It is also important to ensure that the limits of integration are properly adjusted when using u substitution.
No, u substitution with arctan is only applicable for integrals involving trigonometric functions, specifically those that can be rewritten in terms of arctan. Other types of integrals may require different substitution techniques.