Integral Function and its Properties
Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by φ (x) = a∫x f (t) dt for all x ϵ [a, b] is called the integral function of the function f(x)
Property I: The integral function of an integrable function is always continuous.
Property II: If φ (x) is differentiable on (a, b) and φ'(x) = f(x) for all x ϵ (a, b).
Property III: The integral function of an odd function is an even function. If f(x) is an odd function, then φ (x) = φ (x) = a∫x f (t) dt is a function.
If ∫ f (x) dx = F (x) + c then a∫x f (x) dx = F (b) – F (a)
Example: Find the value of ₒ∫⅔1/ (4 + 9x²) dx
Solution: Given that ₒ∫⅔1/ (4 + 9x²) dx
1/9 ₒ∫⅔1/ [(4/9) + (9/9)x²]dx
1/9 ₒ∫⅔1/ [(4/9) + x²]dx
1/9 ₒ∫⅔1/ [(2/3)² + x²]dx
[∵ ∫ dx/ (a² + x²) = 1/a tan⁻¹ (x/a)]
= 1/9. (⅟ ⅔) [tan⁻¹ (x/ ⅔)]ₒ⅔
= ⅙ [tan⁻¹ (3x/2)]ₒ⅔
We can apply boundaries.
= ⅙ [tan⁻¹ (3(⅔)/2)] – ⅙ [tan⁻¹ (3(0)/2)]
= ⅙ [tan⁻¹ (1)] – ⅙ [tan⁻¹ (0)]
= ⅙ x π/4 = π/ 24