# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.5 - Integration [Latest edition]

## Solutions for Chapter 1.5: Integration

Below listed, you can find solutions for Chapter 1.5 of Maharashtra State Board SCERT Maharashtra Question Bank for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022.

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Q.1

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.5 Integration Q.1

#### MCQ [1 Mark]

Q.1 | Q 1

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

Q.1 | Q 2

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"

Q.1 | Q 3

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"

Q.1 | Q 4

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

Q.1 | Q 5

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

Q.1 | Q 6

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

Q.1 | Q 7

Choose the correct alternative:

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

Q.1 | Q 8

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"

Q.1 | Q 9

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

Q.1 | Q 10

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

Q.2

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.5 Integration Q.2

#### Fill in the blanks [1 Mark]

Q.2 | Q 1

int 1/x  "d"x = ______ + c

Q.2 | Q 2

int 1/(x^2 - "a"^2)  "d"x = ______ + c

Q.2 | Q 3

int (7x + 9)^13  "d"x ______ + c

Q.2 | Q 4

int"e"^(4x - 3) "d"x = ______ + c

Q.2 | Q 5

int 5^(6x + 9) "d"x = ______ + c

Q.2 | Q 6

int (5(x^6 + 1))/(x^2 + 1) "d"x = x5 – ______ x3 + 5x + c

Q.2 | Q 7

int (x^2 + x - 6)/((x - 2)(x - 1))  "d"x = x + ______ + c

Q.2 | Q 8

If f'(x) = 1/x + x and f(1) = 5/2, then f(x) = log x + x^2/2 + ______ + c

Q.2 | Q 9

To find the value of int ((1 + logx))/x dx the proper substitution is ______

Q.2 | Q 10

int 1/x^3 [log x^x]^2  "d"x = p(log x)3 + c Then p = ______

Q.3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.5 Integration Q.3

#### [1 Mark]

Q.3 | Q 1

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

Q.3 | Q 2

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

Q.3 | Q 3

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

Q.3 | Q 4

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

Q.3 | Q 5

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

Q.3 | Q 6

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

Q.3 | Q 7

If f(x) = k, where k is constant, then int "k"  "d"x = 0

• True

• False

Q.3 | Q 8

State whether the following statement is True or False:

int3^(2x + 3)  "d"x = (3^(2x + 3))/2 + "c"

• True

• False

Q.3 | Q 9

State whether the following statement is True or False:

int"e"^(4x - 7)  "d"x = ("e"^(4x - 7))/(-7) + "c"

• True

• False

Q.3 | Q 10

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

Q.4

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.5 Integration Q.4

#### Solve the following [3 Marks]

Q.4 | Q 1

Evaluate int(3x^2 - 5)^2  "d"x

Q.4 | Q 2

Evaluate int 1/(x(x - 1))  "d"x

Q.4 | Q 3

Evaluate int 1/(x log x)  "d"x

Q.4 | Q 4

Evaluate int (2"e"^x + 5)/(2"e"^x + 1)  "d"x

Q.4 | Q 5

Evaluate int 1/(4x^2 - 1)  "d"x

Q.4 | Q 6

Evaluate  int"e"^x (1/x - 1/x^2)  "d"x

Q.4 | Q 7

Evaluate int (2x + 1)/((x + 1)(x - 2))  "d"x

Q.4 | Q 8

Evaluate int x log x  "d"x

Q.4 | Q 9

Evaluate int x^2"e"^(4x)  "d"x

Q.4 | Q 10

Evaluate the following.

int 1/(sqrt(3"x"^2 + 8)) dx

Q.5

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.5 Integration Q.5

#### Evaluate the following [4 Marks]

Q.5 | Q 1

int "e"^x x/(x + 1)^2  "d"x

Q.5 | Q 2

int x^3"e"^(x^2) "d"x

Q.5 | Q 3

int x/((x - 1)^2 (x + 2)) "d"x

Q.5 | Q 4

int logx/(1 + logx)^2  "d"x

Q.5 | Q 5

int 1/sqrt(x^2 - 8x - 20)  "d"x

Q.5 | Q 6

int 1/(4x^2 - 20x + 17)  "d"x

Q.5 | Q 7

int (1 + x)/(x + "e"^(-x))  "d"x

Q.5 | Q 8

Evaluate the following.

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

Q.5 | Q 9

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

Q.5 | Q 10

If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).

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## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.5 - Integration

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