Learn definition properties of definite integrals. Practice example using the properties of definite integrals.

(1) \(\displaystyle\int\limits^b_af(x)dx=\,–\displaystyle\int\limits^a_bf(x)dx\)

If we interchange the limits the integral becomes negative of itself.

The value of \(\Delta x=\dfrac{b–a}{n}\) changes sign as \(\dfrac{b–a}{n}\) becomes \(\dfrac{a–b}{n}\).

e.g. \(\displaystyle\int\limits^3_2\dfrac{1}{x^2}dx=\,–\displaystyle\int\limits^2_3\dfrac{1}{x^2}dx\)

(2) \(\displaystyle\int\limits^a_af(x)dx=0\to\) If upper and lower limits are same the value is 0.

A 8

B \(\dfrac{7}{2}\)

C \(\dfrac{5}{2}\)

D \(\dfrac{–2}{3}\)

\(\displaystyle\int\limits^{b}_{a}c\,dx=c\,(b–a)\) = area of rectangle whose height is \('c'\) and width is (b – a).

\(\displaystyle\int\limits^{b}_{a}c\,f(x)dx=c\int\limits^{b}_{a}f(x)dx\)

Where \(c\) is constant and does not depend on \(x\).

\(\displaystyle\underbrace{\int\limits^{b}_{a}f(x)dx}_{\text{Area (1)}}+\underbrace{\int\limits^{c}_{b}f(x)dx}_{\text{Area (2)}}\,=\displaystyle\underbrace{\int\limits^{c}_{a}f(x)dx}_{\text{Area (1)+(2)}}\) ...(1)

- Can also be written as \(\displaystyle\int\limits^{b}_{a}f(x)dx=\int\limits^{c}_{a}f(x)dx\,–\displaystyle\int\limits^{c}_{b}f(x)dx\)

\(\displaystyle\int\limits^{b}_{a}\left(f(x)+g(x)\right)dx\)\(=\displaystyle\left(\int\limits^{b}_{a}f(x)dx\right)+\left(\int\limits^{b}_{a}g(x)dx\right)\)

(The integral of a sum is the sum of integrals.)

A –18

B 16

C 24

D \(\dfrac{1}{4}\)

\(\displaystyle\int\limits^{b}_{a}\left(f(x)–g(x)\right)dx\) \(=\displaystyle\int\limits^{b}_{a}f(x)dx\,–\int\limits^{b}_{a}g(x)dx\)

(The integral of difference is equal to the difference of integrals.)