# Integral Function and its Properties

## 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) = ax 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) = ax f (t) dt is a function.

If ∫ f (x) dx = F (x) + c then ax f (x) dx = F (b) – F (a)

Example: Find the value of ₒ∫1/ (4 + 9x²) dx

Solution: Given that ₒ∫1/ (4 + 9x²) dx

1/9 ₒ∫1/ [(4/9) + (9/9)x²]dx

1/9 ₒ∫1/ [(4/9) + x²]dx

1/9 ₒ∫1/ [(2/3)² + x²]dx

[∵ ∫ dx/ (a² + x²) = 1/a tan⁻¹ (x/a)]

= 1/9. (⅟ ⅔) [tan⁻¹ (x/ ⅔)]ₒ

= ⅙ [tan⁻¹ (3x/2)]ₒ

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