What is the Integration Strategy for Arctan (sqrt(x)) and Arcsin (sqrt(x))?

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SUMMARY

The integration strategy for arctan(sqrt(x)) and arcsin(sqrt(x)) involves using integration by parts. For arctan(sqrt(x)), set u = arctan(sqrt(x)) and dv = dx, leading to du = dx / (1 + (sqrt(x))^2) and v = x. A substitution is necessary to simplify the integration process further. This method effectively addresses both functions through systematic application of integration techniques.

PREREQUISITES
  • Understanding of integration by parts
  • Familiarity with the arctangent and arcsine functions
  • Knowledge of substitution methods in calculus
  • Basic proficiency in handling square root functions
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  • Study advanced integration techniques, focusing on integration by parts
  • Explore substitution methods in calculus for complex functions
  • Learn about the properties and applications of arctan and arcsin functions
  • Practice integrating similar functions to reinforce understanding
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Students and educators in calculus, mathematicians focusing on integration techniques, and anyone seeking to deepen their understanding of trigonometric integrals.

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What is a strategy to integrate arctan (sqrt(x))?
What about arcsin (sqrt(x))?
 
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use integration by parts. Let u=arctan(...) and dv=dx, so du=dx/[1+(...)^2], v=x


then on the next line you would have to use a substitution.
 

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