# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.6 - Definite Integration [Latest edition]

## Solutions for Chapter 1.6: Definite Integration

Below listed, you can find solutions for Chapter 1.6 of Maharashtra State Board SCERT Maharashtra Question Bank for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022.

Q.1Q.2Q.3Q.4Q.5Q.6
Q.1

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.1

#### MCQ [1 Mark]

Q.1 | Q 1

Choose the correct alternative:

int_2^3 x^4  "d"x =

• 1/2

• 5/2

• 5/211

• 211/5

Q.1 | Q 2

Choose the correct alternative:

int_0^"a" 3x^5  "d"x = 8, then a =

• 2

• 0

• 8/3

• a

Q.1 | Q 3

Choose the correct alternative:

int_4^9 ("d"x)/sqrt(x) =

• 9

• 4

• 2

• 0

Q.1 | Q 4

int_0^2 e^x dx = ______.

• e2 – 1

• 1 – e2

• e – 1

• 1 – e

Q.1 | Q 5

Choose the correct alternative:

int_(-2)^3 1/(x + 5)  "d"x =

• log (3/8)

• log (8/3)

• log (8/5)

• 0

Q.1 | Q 6

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)

Q.1 | Q 7

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

Q.1 | Q 8

int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))  "d"x = ______

• 7/2

• 5/2

• 7

• 2

Q.1 | Q 9

Choose the correct alternative:

int_(-9)^9 x^3/(4 - x^2)  "d"x =

• 0

• 3

• 9

• – 9

Q.2

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.2

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

Q.2 | Q 1

int_1^2 x^2  "d"x = ______

Q.2 | Q 2

int_0^"a" 4x^3  "d"x = 81, then a = ______

Q.2 | Q 3

int_0^1 "e"^(2x) "d"x = ______

Q.2 | Q 4

int_1^2 1/(2x + 3)  dx = ______

Q.2 | Q 5

int_2^4 x/(x^2 + 1)  "d"x = ______

Q.2 | Q 6

int_(-7)^7 x^3/(x^2 + 7)  "d"x = ______

Q.3

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.3

#### [1 Mark]

Q.3 | Q 1

State whether the following statement is True or False:

int_0^"a" 3x^2  "d"x = 27, then a = 2.5

• True

• False

Q.3 | Q 2

State whether the following statement is True or False:

int_0^1 1/(2x + 5)  "d"x = log(7/5)

• True

• False

Q.3 | Q 3

State whether the following statement is True or False:

int_2^3 x/(x^2 + 1)  "d"x = 1/2 log 2

• True

• False

Q.3 | Q 4

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

Q.3 | Q 5

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

Q.3 | Q 6

State whether the following statement is True or False:

int_(-5)^5 x/(x^2 + 7)  "d"x = 10

• True

• False

Q.4

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.4

#### Solve the following [3 Marks]

Q.4 | Q 1

Evaluate int_0^1 (x^2 + 3x + 2)/sqrt(x)  "d"x

Q.4 | Q 2

Evaluate int_0^1 1/(sqrt(1 + x) + sqrt(x))  "d"x

Q.4 | Q 3

If int_0^"a" (2x + 1)  "d"x = 2, find a

Q.4 | Q 4

If int_1^"a" (3x^2 + 2x + 1)  "d"x = 11, find the real value of a

Q.4 | Q 5

Evaluate int_1^"e" 1/(x(1 + log x)^2)  "d"x

Q.4 | Q 6

Evaluate int_1^2 1/(x^2 + 6x + 5)  "d"x

Q.4 | Q 7

Evaluate int_1^2 (3x)/((9x^2 - 1))  "d"x

Q.4 | Q 8

Evaluate the following integrals:

int_1^3 (root(3)(x + 5))/(root(3)(x + 5) + root(3)(9 - x))*dx

Q.4 | Q 9

Evaluate int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x))  "d"x

Q.4 | Q 10

Evaluate the following integrals : int_0^1 log(1/x - 1)*dx

Q.5

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.5

#### [4 Marks]

Q.5 | Q 1

Evaluate int_2^3 x/((x + 2)(x + 3))  "d"x

Q.5 | Q 2

Evaluate int_1^2 "e"^(2x) (1/x - 1/(2x^2))  "d"x

Q.5 | Q 3

Evaluate int_1^3 log x  "d"x

Q.5 | Q 4

Evaluate int_1^3 x^2*log x  "d"x

Q.5 | Q 5

Evaluate int_0^1 "e"^(x^2)*"x"^3  "d"x

Q.5 | Q 6

Evaluate int_0^"a" x^2 ("a" - x)^(3/2)  "d"x

Q.5 | Q 7

Evaluate int_0^1 x(1 - x)^5  "d"x

Q.6

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 1.6 Definite Integration Q.6

#### Activity [4 Marks]

Q.6 | Q 1

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

Q.6 | Q 2

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 Chapter 1.6: Definite Integration

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## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.6 - Definite Integration

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 1.6 Definite Integration are Fundamental Theorem of Integral Calculus, Properties of Definite Integrals.

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