5.7 Integrals Resulting in Inverse Trigonometric Functions
2 min read•june 24, 2024
are key players in calculus. They pop up when we integrate certain tricky expressions, giving us a way to solve problems that might otherwise stump us.
Knowing how to spot and handle these integrals is crucial. We'll learn to recognize standard forms, use substitution techniques, and navigate domain restrictions. These skills will help us tackle a wide of integration challenges.
Integrals Resulting in Inverse Trigonometric Functions
Inverse trigonometric function integrals
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Recognize the standard forms of integrals that yield
∫a2−x21dx=sin−1(ax)+C for ∣x∣<∣a∣ (arcsine)
∫a2+x21dx=a1tan−1(ax)+C (arctangent)
∫xx2−a21dx=a1sec−1(a∣x∣)+C for ∣x∣>∣a∣ (arcsecant)
Identify the appropriate formula based on the integrand's structure
Look for the presence of a2−x2, a2+x2, or xx2−a2 in the denominator
Evaluate the integral by applying the corresponding formula
Simplify the resulting expression and include the arbitrary constant of integration (C)
Substitution for inverse trigonometric forms
Recognize integrals that can be transformed into one of the standard inverse trigonometric forms
Apply trigonometric substitution to convert the integrand into a recognizable form
For a2−x2, substitute x=asinθ and dx=acosθdθ
For x2+a2, substitute x=atanθ and dx=asec2θdθ
For x2−a2, substitute x=asecθ and dx=asecθtanθdθ
Express the integral in terms of the new variable (θ)
Replace x and dx with their respective substitutions
Solve the simplified integral using the appropriate inverse trigonometric function formula
Substitute back the original variable (x) and simplify the result
Replace θ with its equivalent expression in terms of x
Use u-substitution when appropriate to simplify complex integrands
Domains and ranges in integration
Understand the domain restrictions for each inverse trigonometric function resulting from integration
sin−1(x): Domain [−1,1], Range [−2π,2π]
tan−1(x): Domain (−∞,∞), Range (−2π,2π)
sec−1(x): Domain (−∞,−1]∪[1,∞), Range [0,π] (excluding 2π)
Check that the resulting expression satisfies the domain restrictions
Ensure ∣x∣<∣a∣ for sin−1(ax)
No domain restriction for tan−1(ax)
Ensure ∣x∣>∣a∣ for sec−1(a∣x∣)
Adjust the constant of integration (C) if needed to ensure the result falls within the appropriate range
Add or subtract multiples of π to shift the range as necessary
Advanced integration techniques
Apply partial fractions decomposition for complex rational functions before integration
Use integration by parts for products of functions involving inverse trigonometric expressions
Utilize trigonometric identities to simplify integrands when necessary
Remember to apply the chain rule when dealing with composite functions in inverse trigonometric integrals