Introduction
Integrating the function sec(tan^-1(x)) can be approached using trigonometric identities and substitution techniques. Let’s break down the steps in detail.
Step 1: Understanding the Function
- The expression sec(tan^-1(x)) is the secant of the angle whose tangent is x.
- By definition, if θ = tan^-1(x), then tan(θ) = x.
Step 2: Using Right Triangle Relationships
- Create a right triangle:
- Opposite side = x
- Adjacent side = 1
- Hypotenuse = √(x^2 + 1)
- Therefore, sec(θ) = Hypotenuse / Adjacent = √(x^2 + 1) / 1 = √(x^2 + 1).
Step 3: Setting Up the Integral
- The integral we need to evaluate becomes:
∫ sec(tan^-1(x)) dx = ∫ √(x^2 + 1) dx.
Step 4: Performing the Integration
- The integral ∫ √(x^2 + 1) dx can be solved using the substitution method or recognizing it as a standard form.
- The antiderivative is:
(1/2)(x√(x^2 + 1) + ln|x + √(x^2 + 1)|) + C.
Step 5: Conclusion
- The final result of the integral ∫ sec(tan^-1(x)) dx is:
(1/2)(x√(x^2 + 1) + ln|x + √(x^2 + 1)|) + C.
This structured approach simplifies the integration of sec(tan^-1(x)), leveraging trigonometric identities and geometrical interpretations.