Newton-Leibnitz Formula, Integral Function and its Properties

Newton-Leibnitz Formula, Integral Function and its Properties

Let f(x) be a continuous function defined on [a, b], then a function ф(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)x

Property I: The integral function of an integrable function is always continuous.

Property II: If ф(x) is differentiable on (a, b) and ф’(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 \(\left[ \frac{5\pi }{3},\frac{7\pi }{4} \right]\)

Solution:  We have

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

⇒ F’(x) = 6cosX – 2sinx

For all \(x\in \left[ 5\pi /3,7\pi /4 \right]\), we have

cosx > 0 and sinx < 0

⇒ F’(x) = 6cosx – 2sinx >0

⇒ F(x) is an increasing function \(\left[ \frac{5\pi }{3},\frac{7\pi }{4} \right]\)

⇒ F(x) is greatest value x = \(\frac{7\pi }{4}\)


Greatest value = \(F\left( \frac{7\pi }{4} \right)\)

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

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


The Newton-Leibnitz formula or the fundamental theorem of integral calculus

If ф(x) is one of the primitives or antiderivatives of a function f(x) defined on [a, b], then the define integral of f(x) over [a, b] is given by

ф(b) – ф(a) and is denoted by \(\int_{a}^{b}{f\left( x \right)dx.}\).

Thus, if \(\frac{d}{dx}\left( \phi \left( x \right) \right)=f\left( x \right)\), then \(\int_{a}^{b}{f\left( x \right)dx}=\phi \left( b \right)-\phi \left( a \right)\).

The numbers a and b are called the limits of integration ‘a’ is called the lower limit ‘b’ is the upper limit. The interval [a, b] is called the interval of integration.