Integral is it Partial Fraction

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SUMMARY

The integral \(\int_0^∞ \frac{dv}{(1+v^2)(1+\arctan v)}\) requires the application of substitution techniques for evaluation. The discussion emphasizes using the substitution \(z = \arctan(v)\) to transform the integral, which simplifies the integration process. Participants noted the importance of correctly determining \(dz\) when substituting variables to ensure the integral is expressed in the new variable. This approach is essential for handling integrals involving inverse trigonometric functions in the denominator.

PREREQUISITES
  • Understanding of integral calculus, specifically improper integrals.
  • Familiarity with inverse trigonometric functions, particularly arctangent.
  • Knowledge of substitution methods in integration.
  • Basic proficiency in handling partial fractions.
NEXT STEPS
  • Study the method of substitution in integrals, focusing on inverse trigonometric functions.
  • Learn how to derive \(dz\) when substituting variables in integrals.
  • Explore techniques for integrating rational functions using partial fractions.
  • Practice evaluating improper integrals with complex denominators.
USEFUL FOR

Students and educators in calculus, mathematicians dealing with integral evaluations, and anyone seeking to enhance their skills in integration techniques involving inverse trigonometric functions.

whatlifeforme
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Homework Statement


Evaluate the integral.

Homework Equations


[itex]\displaystyle\int_0^∞ {\frac{dv}{(1+v^2)(1+arctanv)}}[/itex]


The Attempt at a Solution


i am not sure how to do partial fractions with an inverse trig function in the denominator.

i tried v=arctanv so i could do a substitution, but that would be correct, and i don't know how using another variable would work such as z=arctanv.
 
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The letter you use for your substitution does not have to be any special letter, as long as it doesn't confuse you. If you feel like it, write arctan(v) as z. However we now need to write the entire integral in terms of z and we need dz if we want to integrate it in this form. If z = arctan(v), then what is dz ? Is dz present in the integral ?
 
whatlifeforme said:

Homework Statement


Evaluate the integral.

Homework Equations


[itex]\displaystyle\int_0^∞ {\frac{dv}{(1+v^2)(1+arctanv)}}[/itex]


The Attempt at a Solution


i am not sure how to do partial fractions with an inverse trig function in the denominator.

i tried v=arctanv so i could do a substitution, but that would be correct, and i don't know how using another variable would work such as z=arctanv.

v=arctan(v) is nonsense. z=arctan(v) should work. Try it.
 

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