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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.

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

A 28

B 36

C –18

D $$\dfrac{1}{6}$$

×

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

$$=3(7+5)$$

$$=36$$

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 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 $$x$$.

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

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

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.)  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.)  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