Properties and Formulas for Definite Integrals

Here, you will learn formulas for definite integrals and properties of definite integrals with examples.

Let’s begin –

A definite integral is denoted by $\int_{a}^{b}$ f(x)dx which represent the algebraic area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis.

Properties and Formulas for Definite Integrals

(a)  $\int_{a}^{b}$ f(x)dx = $\int_{a}^{b}$ f(t)dt provided f is same

(b)  $\int_{a}^{b}$ f(x)dx = – $\int_{b}^{a}$ f(x)dx

(c)  $\int_{a}^{b}$ f(x)dx = $\int_{a}^{c}$ f(x)dx + $\int_{c}^{b}$ f(x)dx , where c may lie inside or outside the interval [a,b]. This property is to be used when f is piecewise continous in (a, b).

(d)  $\int_{a}^{a}$ f(x)dx = $\int_{0}^{a}$ [f(x) + f(-x)]dx = $\begin{cases} 0 & \text{if f(x) is an odd function}\ \\ 2\int_{a}^{b} f(x)dx & \text{if f(x) is an even function}\ \end{cases}$

Example : Evaluate $\int_{1/2}^{1/2}$ $cosx ln{({1+x\over 1-x})}$ dx

Solution : f(-x) = $cos(-x) ln{({1-x\over 1+x})}$ = – $cosx ln{({1+x\over 1-x})}$ = f(-x)

$\implies$   f(x) is odd

Hence, the value of the given interval is 0.

(e)  $\int_{a}^{b}$ f(x)dx = $\int_{a}^{b}$ f(a+b-x)dx, In particular $\int_{0}^{a}$ f(x)dx = $\int_{0}^{a}$ f(a-x)dx

Example : Evaluate $\int_{0}^{\pi/2}$ $asinx+bcosx\over sinx+cosx$ dx

Solution : I = $\int_{0}^{\pi/2}$ $asinx+bcosx\over sinx+cosx$ dx     ….(i)

I = $\int_{0}^{\pi/2}$ $asin(\pi/2-x)+bcos(\pi/2-x)\over sin(\pi/2-x)+cos(\pi/2-x)$ dx = $\int_{0}^{\pi/2}$ $acosx+bsinx\over sinx+cosx$ dx     ….(ii)

Adding (i) and (ii),

2I = $\int_{0}^{\pi/2}$ $a+b)(sinx+cosx)\over sinx+cosx$ dx = $\int_{0}^{\pi/2}$ (a+b) dx = (a+b)$\pi/2$

$\implies$   I = (a+b)$\pi/4$

(f)  $\int_{0}^{2a}$ f(x)dx = $\int_{0}^{a}$ f(x)dx + $\int_{0}^{a}$ f(2a-x)dx = $\begin{cases} 2\int_{0}^{a} f(x)dx & \text{if}\ f(2a-x) = f(x) \\ 0 & \text{if}\ f(2a-x) = -f(x) \end{cases}$.

(g)  $\int_{0}^{nT}$ f(x)dx = n$\int_{0}^{T}$ f(x)dx, (n $\in$ I); where T is the period of the function i.e. f(T+x) = f(x)

(h)  $\int_{a+nT}^{b+nT}$ f(x)dx = $\int_{a}^{b}$ f(x)dx, where f(x) is periodic with period T & n $\in$ I.

(i)  $\int_{mT}^{nT}$ f(x)dx = (n-m)$\int_{0}^{T}$ f(x)dx, where f(x) is periodic with period T & (n, m $\in$ I).