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).
A 28
B 36
C –18
D \(\dfrac{1}{6}\)
\(\displaystyle\int\limits^{b}_{a}c\,dx=c\,(b–a)\)
\(\displaystyle\int\limits^{7}_{–5}3\,dx=3\,\left(7–(–5)\right)\)
\(=3(7+5)\)
\(=36\)
Option B is Correct
\(\displaystyle\int\limits^{b}_{a}c\,f(x)dx=c\int\limits^{b}_{a}f(x)dx\)
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)
\(\displaystyle\int\limits^{b}_{a}\left(f(x)+g(x)\right)dx\)\(=\displaystyle\left(\int\limits^{b}_{a}f(x)dx\right)+\left(\int\limits^{b}_{a}g(x)dx\right)\)
(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.)
