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Chapters
1.2: Matrices
1.3: Differentiation
1.4: Applications of Derivatives
1.5: Integration
▶ 1.6: Definite Integration
1.7: Application of Definite Integration
1.8: Differential Equation and Applications
2.1: Commission, Brokerage and Discount
2.2: Insurance and Annuity
2.3: Linear Regression
2.4: Time Series
2.5: Index Numbers
2.6: Linear Programming
2.7: Assignment Problem and Sequencing
2.8: Probability Distributions
![SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.6 - Definite Integration - Shaalaa.com](/images/mathematics-and-statistics-commerce-english-12-standard-hsc_6:5f2b1b2038084cf381bfa42c826a928c.jpg "SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.6 - Definite Integration")
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Solutions for Chapter 1.6: Definite Integration
Below listed, you can find solutions for Chapter 1.6 of Maharashtra State Board SCERT Maharashtra for Mathematics and Statistics (Commerce) [English] 12 Standard HSC.
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.6 Definite Integration Q.1
MCQ [1 Mark]
Choose the correct alternative:
`int_2^3 x^4 "d"x` =
`1/2`
`5/2`
`5/211`
`211/5`
Choose the correct alternative:
`int_0^"a" 3x^5 "d"x` = 8, then a =
2
0
`8/3`
a
Choose the correct alternative:
`int_4^9 ("d"x)/sqrt(x)` =
9
4
2
0
`int_0^2 e^x dx` = ______.
e2 – 1
1 – e2
e – 1
1 – e
Choose the correct alternative:
`int_(-2)^3 1/(x + 5) "d"x` =
`log (3/8)`
`log (8/3)`
`log (8/5)`
0
Choose the correct alternative:
`int_2^3 x/(x^2 - 1) "d"x` =
`log (8/3)`
`- log (8/3)`
`1/2 log(8/3)`
`-1/2 log(8/3)`
`int_"a"^"b" "f"(x) "d"x` = ______
`int_"b"^"a" "f"(x) "d"x`
`- int_"a"^"b" "f"(x) "d"x`
`- int_"b"^"a" "f"(x) "d"x`
`int_0^"a" "f"(x) "d"x`
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
`7/2`
`5/2`
7
2
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
0
3
9
– 9
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.6 Definite Integration Q.2
Fill in the blanks [1 Mark]
`int_1^2 x^2 "d"x` = ______
`int_0^"a" 4x^3 "d"x` = 81, then a = ______
`int_0^1 "e"^(2x) "d"x` = ______
`int_1^2 1/(2x + 3) dx` = ______
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.6 Definite Integration Q.3
[1 Mark]
State whether the following statement is True or False:
`int_0^"a" 3x^2 "d"x` = 27, then a = 2.5
True
False
State whether the following statement is True or False:
`int_0^1 1/(2x + 5) "d"x = log(7/5)`
True
False
State whether the following statement is True or False:
`int_2^3 x/(x^2 + 1) "d"x = 1/2 log 2`
True
False
State whether the following statement is True or False:
`int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
True
False
State whether the following statement is True or False:
`int_0^(2"a") "f"(x) "d"x = int_0^"a" "f"(x) "d"x + int_0^"a" "f"("a" - x) "d"x`
True
False
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
True
False
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.6 Definite Integration Q.4
Solve the following [3 Marks]
Evaluate `int_0^1 (x^2 + 3x + 2)/sqrt(x) "d"x`
Evaluate `int_0^1 1/(sqrt(1 + x) + sqrt(x)) "d"x`
If `int_0^"a" (2x + 1) "d"x` = 2, find a
If `int_1^"a" (3x^2 + 2x + 1) "d"x` = 11, find the real value of a
Evaluate `int_1^"e" 1/(x(1 + log x)^2) "d"x`
Evaluate:
`int_1^2 1/(x^2 + 6x + 5) dx`
Evaluate `int_1^2 (3x)/((9x^2 - 1)) "d"x`
Evaluate the following integrals:
`int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx`
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
Evaluate the following integrals : `int_0^1 log(1/x - 1)*dx`
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.6 Definite Integration Q.5
[4 Marks]
Evaluate `int_2^3 x/((x + 2)(x + 3)) "d"x`
Evaluate `int_1^2 "e"^(2x) (1/x - 1/(2x^2)) "d"x`
Evaluate `int_1^3 log x "d"x`
Evaluate `int_1^3 x^2*log x "d"x`
Evaluate `int_0^1 "e"^(x^2)*"x"^3 "d"x`
Evaluate `int_0^"a" x^2 ("a" - x)^(3/2) "d"x`
Evaluate `int_0^1 x(1 - x)^5 "d"x`
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.6 Definite Integration Q.6
Activity [4 Marks]
By completing the following activity, Evaluate `int_1^2 (x + 3)/(x(x + 2)) "d"x`
Solution: Let I = `int_1^2 (x + 3)/(x(x + 2)) "d"x`
Let `(x + 3)/(x(x + 2)) = "A"/x + "B"/((x + 2))`
∴ x + 3 = A(x + 2) + B.x
∴ A = `square`, B = `square`
∴ I = `int_1^2[("( )")/x + ("( )")/((x + 2))] "d"x`
∴ I = `[square log x + square log(x + 2)]_1^2`
∴ I = `square`
By completing the following activity, Evaluate `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x`.
Solution: Let I = `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x` ......(i)
Using the property, `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`, we get
I = `int_2^5 ("( )")/(sqrt(7 - x) + "( )") "d"x` ......(ii)
Adding equations (i) and (ii), we get
2I = `int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x)) "d"x + ( ) "d"x`
2I = `int_2^5 (("( )" + "( )")/("( )" + "( )")) "d"x`
2I = `square`
∴ I = `square`
Solutions for 1.6: Definite Integration
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.6 - Definite Integration
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Concepts covered in Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.6 Definite Integration are Fundamental Theorem of Integral Calculus, Properties of Definite Integrals.
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