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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 1.5 - 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 1.5 - Integration")
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Solutions for Chapter 1.5: Integration
Below listed, you can find solutions for Chapter 1.5 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 1.5 Integration Q.1
MCQ [1 Mark]
Choose the correct alternative:
The value of `int ("d"x)/sqrt(1 - x)` is
`2sqrt(1 - x) + "c"`
`-2sqrt(1 - x) + "c"`
`sqrt(x) + "c"`
x + c
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`x/2 sqrt(1 + x) + "c"`
`2/3(1 + x)^(3/2) + "c"`
`2/sqrt(1 + x) + "c"`
`(-3)/2 (1 + x) + "c"`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`(3)^(x^3) + "c"`
`((3)^(x^3))/(3log3) + "c"`
`log 3*(3)^(x^3) + "c"`
`x^2 (3)^(x^2) + "c"`
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
`1/3`
`1/2`
`1/4`
2
`int ("d"x)/(x - x^2)` = ______
log x – log(1 – x) + c
log(1 – x2) + c
– log x + log(1 – x) + c
log(x – x2) + c
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`1/15 log((x + 2)/(x - 1)) + "c"`
`1/15 log((x + 8)/(x + 7)) + "c"`
`1/15 log((x - 8)/(x + 7)) + "c"`
(x – 8)(x – 7) + c
`int(x + 1/x)^3 dx` = ______.
`1/4(x + 1/x)^4 + c`
`x^4/4 + (3x^2)/2 + 3log x - 1/(2x^2) + c`
`x^4/4 + (3x^2)/2 + 3log x + 1/x^2 + c`
`(x - x^(-1))^3 + c`
Choose the correct alternative:
`int(("e"^(2x) + "e"^(-2x))/"e"^x) "d"x` =
`"e"^x - 1/(3"e"^(3x)) + "c"`
`"e"^x + 1/(3"e"^(3x)) + "c"`
`"e"^(-x) - 1/(3"e"^(3x)) + "c"`
`"e"^(-x) + 1/(3"e"^(3x)) + "c"`
Choose the correct alternative:
`int(1 - x)^(-2) dx` = ______.
`(1 - x)^(-1) + c`
`(1 + x)^(-1) + c`
`(1 - x)^(-1) - 1 + c`
`(1 - x)^(-1) + 1 + c`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
`(-1)/(x + 1) + "c"`
`((-1)/(x + 1))^5 + "c"`
log(x + 1) + c
5log(x + 5) + c
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.5 Integration Q.2
Fill in the blanks [1 Mark]
`int 1/x "d"x` = ______ + c
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (7x + 9)^13 "d"x` ______ + c
`int"e"^(4x - 3) "d"x` = ______ + c
`int 5^(6x + 9) "d"x` = ______ + c
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.5 Integration Q.3
[1 Mark]
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
True
False
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
True
False
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
True
False
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
True
False
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
True
False
State whether the following statement is True or False:
If f'(x) = 3x2 + 2x, then by definition of integration, we get f(x) = x3 + x2 + c
True
False
If f(x) = k, where k is constant, then `int "k" "d"x` = 0
True
False
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
True
False
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
True
False
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
True
False
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.5 Integration Q.4
Solve the following [3 Marks]
Evaluate `int(3x^2 - 5)^2 "d"x`
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate `int 1/(x log x) "d"x`
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int 1/(4x^2 - 1) "d"x`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Evaluate `int x log x "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
Evaluate the following.
`int 1/(sqrt(3"x"^2 + 8))` dx
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC 1.5 Integration Q.5
Evaluate the following [4 Marks]
`int "e"^x x/(x + 1)^2 "d"x`
`int x^3"e"^(x^2) "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
`int logx/(1 + logx)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int (1 + x)/(x + "e"^(-x)) "d"x`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Solutions for 1.5: Integration
SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.5 - Integration
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Concepts covered in Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 1.5 Integration are Integration, Methods of Integration: Integration by Substitution, Methods of Integration: Integration by Parts, Methods of Integration: Integration Using Partial Fractions.
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