Integrals Resulting in Inverse Trigonometric Functions: Fresh Take

  • Calculate integrals that lead to inverse trigonometric function solutions

Integrals Resulting in Inverse Trigonometric Functions

The Main Idea 

  • Integration formulas yielding inverse trigonometric functions
  • Domain restrictions for inverse trigonometric functions
  • Connection between derivatives of inverse trig functions and these integrals
  • Application of these formulas to various types of integrals
  • Handling negative integrands in inverse trig integrals

Key Formulas

  • dua2u2=sin1(u|a|)+C
  • dua2+u2=1atan1(ua)+C
  • duuu2a2=1|a|sec1(|u|a)+C

Key Concepts

  • Domain Restrictions:
    • Inverse trig functions have restricted domains
    • Solutions must respect these domain restrictions
  • Recognizing Integrands:
    • Identify integrands that match standard forms
    • Use substitution to transform integrals into these forms
  • Relationship to Derivatives:
    • These integrals are related to derivatives of inverse trig functions
    • Understanding this connection aids in application
  • Handling Negative Integrands:
    • Factor out 1 from negative integrands
    • Use existing formulas rather than memorizing new ones
  • Substitution Techniques:
    • Often necessary to transform integrals into standard forms
    • May involve adjusting constants or variables

Find the antiderivative of dx116x2.

Find the indefinite integral using an inverse trigonometric function and substitution for dx9x2.

Use substitution to find the antiderivative of dx25+4x2.

Find the antiderivative of dx16+x2.

Evaluate the definite integral 02dx4+x2.