#### Online Mock Tests

#### Chapters

Chapter 1.2: Matrices

Chapter 1.3: Differentiation

Chapter 1.4: Applications of Derivatives

Chapter 1.5: Integration

Chapter 1.6: Definite Integration

Chapter 1.7: Application of Definite Integration

Chapter 1.8: Differential Equation and Applications

Chapter 2.1: Commission, Brokerage and Discount

Chapter 2.2: Insurance and Annuity

Chapter 2.3: Linear Regression

Chapter 2.4: Time Series

Chapter 2.5: Index Numbers

Chapter 2.6: Linear Programming

Chapter 2.7: Assignment Problem and Sequencing

Chapter 2.8: Probability Distributions

## Chapter 2: Integration

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 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

**Choose the correct alternative:**

`int ("d"x)/(x - x^2)` =

log x – log(1 – x) + c

log(1 – x

^{2}) + c– log x + log(1 – x) + c

log(x – x

^{2}) + 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

**Choose the correct alternative:**

`int(x + 1/x)^3 "d"x` =

`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) "d"x` =

`(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 Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 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` = x^{5} – ______ x^{3} + 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 Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 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` = e^{x }f(x) + c, f(x) = (x + 1)^{2}

True

False

**State whether the following statement is True or False:**

If f'(x) = 3x^{2} + 2x, then by definition of integration, we get f(x) = x^{3} + x^{2} + c

True

False

**State whether the following statement is True or 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 Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 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 `int 1/sqrt(3x^2 + 8) "d"x`

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2 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`

`int x^3/sqrt(1 + x^4) "d"x`

`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`

If f'(x) = 4x^{3} – 3x^{2} + 2x + k, f(0) = 1 and f(1) = 4, find f(x)

## Chapter 2: Integration

## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2 - Integration

SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2 (Integration) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2 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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