Integral Function and its Properties

Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by  \(\phi \left( x \right)=\int_{a}^{x}{f\left( t \right)dt}\) 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 \(\phi \left( x \right)=\int_{a}^{x}{f\left( t \right)dt}\) is an function.

Example: Find the greatest value of \(F\left( x \right)=\int_{5\pi /3}^{x}{\left( 6\cos t-2\sin t \right)dt}\) in the interval [5π/3, 7π/4]

Solution: We have,

\(F\left( x \right)=\int_{5\pi /3}^{x}{\left( 6\cos t-2\sin t \right)dt}\).

⇒ F’(x) = 6 cosx – 2sinx

For all x ϵ [5π/3, 7π/4], we have

⇒ cos x > 0 and sin x < 0

⇒ F’(x) = 6 cosx – 2sinx > 0

⇒ F(x) is an increasing function on [5π/3, 7π/4]

⇒ F(x) attains greatest value at x = 7π/4.


Greatest value = F (7π/4)

\(=\int_{5\pi /3}^{7\pi /4}{\left( 6\cos -2\sin t \right)dt}\).

\(=\left[ 6\sin t+2\cos t \right]_{5\pi /3}^{7\pi /4}\).

= 3√3 – 2√2 – 1.