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Properties Of Definite Integral

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

Reversing the Limits and Zero Integral Property 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.

Illustration Questions

For some function \('f'\) if  \(\displaystyle\int\limits^{7}_{–5}f(x)dx=\dfrac{2}{3}\) then the value of \(\displaystyle\int\limits^{–5}_{7}f(x)dx\) is

A 8

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

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

D \(\dfrac{–2}{3}\)

×

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

\(\Rightarrow\,\displaystyle\int\limits^{–5}_{7}f(x)dx=\,–\displaystyle\int\limits^{7}_{–5}f(x)dx=\dfrac{–2}{3}\)

For some function \('f'\) if  \(\displaystyle\int\limits^{7}_{–5}f(x)dx=\dfrac{2}{3}\) then the value of \(\displaystyle\int\limits^{–5}_{7}f(x)dx\) is

A

8

.

B

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

C

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

D

\(\dfrac{–2}{3}\)

Option D is Correct

Property of Definite Integral

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

Illustration Questions

The value of  \(\displaystyle\int\limits^{7}_{–5}3\,dx\) is

A 28

B 36

C –18

D \(\dfrac{1}{6}\)

×

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

image

\(\displaystyle\int\limits^{7}_{–5}3\,dx=3\,\left(7–(–5)\right)\)

\(=3(7+5)\)

\(=36\)

The value of  \(\displaystyle\int\limits^{7}_{–5}3\,dx\) is

A

28

.

B

36

C

–18

D

\(\dfrac{1}{6}\)

Option B is Correct

The Sum Property of Definite Integral

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

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

Illustration Questions

If \(\displaystyle\int\limits^{5}_{2}f(x)dx=7\) then the value of  \(\displaystyle\int\limits^{5}_{2}\left(3+f(x)\right)dx\) is

A –18

B 16

C 24

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

×

\(\displaystyle\int\limits^{5}_{2}\left(3+f(x)\right)dx\)\(=\displaystyle\int\limits^{5}_{2}3\,dx+\int\limits^{5}_{2}f(x)dx\)

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

\(=3(5–2)+7\)

\(=9+7\)

\(=16\)

If \(\displaystyle\int\limits^{5}_{2}f(x)dx=7\) then the value of  \(\displaystyle\int\limits^{5}_{2}\left(3+f(x)\right)dx\) is

A

–18

.

B

16

C

24

D

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

Option B is Correct

The Difference Property of Definite Integrals

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

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

Illustration Questions

If \(\displaystyle\int\limits^{5}_{1}f(x)dx=17\) and  \(\displaystyle\int\limits^{5}_{1}g(x)dx=7\) then the value of \(\displaystyle\int\limits^{5}_{1}\left(f(x)–g(x)\right)dx\) is

A 72

B 10

C –81

D 4.2

×

\(\displaystyle\int\limits^{5}_{1}(f(x)–g(x))dx=\displaystyle\int\limits^{5}_{1}f(x)dx\,–\int\limits^{5}_{1}g(x)dx\)

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

\(=17–7\)

\(=10\)

If \(\displaystyle\int\limits^{5}_{1}f(x)dx=17\) and  \(\displaystyle\int\limits^{5}_{1}g(x)dx=7\) then the value of \(\displaystyle\int\limits^{5}_{1}\left(f(x)–g(x)\right)dx\) is

A

72

.

B

10

C

–81

D

4.2

Option B is Correct

The Constant Multiple Property of Definite Integral(Linearity of Definite Integrals)

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

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

Illustration Questions

If \(\displaystyle\int\limits^{8}_{2}f(x)dx=18\) and \(\displaystyle\int\limits^{8}_{2}g(x)dx=–2\) then find value of \(\displaystyle\int\limits^{8}_{2}\left(5g(x)–3f(x)\right)dx\).

A –64

B 72

C 1

D –5

×

\(\displaystyle\int\limits^{8}_{2}\left(5g(x)–3f(x)\right)dx\) \(=\displaystyle\int\limits^{8}_{2}5g(x)dx\,–\int\limits^{8}_{2}3f(x)dx\)

\(=\displaystyle5\int\limits^{8}_{2}g(x)dx\,–3\int\limits^{8}_{2}f(x)dx\)

\(=5×(–2)\,–3×18\)

\(=–10–54\)

\(=–64\)

If \(\displaystyle\int\limits^{8}_{2}f(x)dx=18\) and \(\displaystyle\int\limits^{8}_{2}g(x)dx=–2\) then find value of \(\displaystyle\int\limits^{8}_{2}\left(5g(x)–3f(x)\right)dx\).

A

–64

.

B

72

C

1

D

–5

Option A is Correct

Additive Interval Property of Definite Integrals

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

 

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

Illustration Questions

If \(\displaystyle\int\limits^{4}_{2}f(x)dx=–11\) and  \(\displaystyle\int\limits^{6}_{2}f(x)dx=5\) then the value of \(\displaystyle\int\limits^{6}_{4}f(x)dx\) equals

A 16

B 82

C –4

D \(\dfrac{1}{5}\)

×

\(\displaystyle\int\limits^{4}_{2}f(x)dx+\int\limits^{6}_{4}f(x)dx\,=\displaystyle\int\limits^{6}_{2}f(x)dx\)

\(\Rightarrow\,\displaystyle–11+\int\limits^{6}_{4}f(x)dx=5\)

\(\Rightarrow\displaystyle\int\limits^{6}_{4}f(x)dx=5+11\)

\(=16\)

If \(\displaystyle\int\limits^{4}_{2}f(x)dx=–11\) and  \(\displaystyle\int\limits^{6}_{2}f(x)dx=5\) then the value of \(\displaystyle\int\limits^{6}_{4}f(x)dx\) equals

A

16

.

B

82

C

–4

D

\(\dfrac{1}{5}\)

Option A is Correct

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