# SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 2 - Definite Integration [Latest edition]

#### Chapters ## Chapter 2: Definite Integration

MCQ

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Definite Integration MCQ

#### 2 Marks each

MCQ | Q 1

int_1^9 (x + 1)/sqrt(x)  "d"x =

• 80/3

• 64/3

• 17/3

• 15/3

MCQ | Q 2

int_0^1 sqrt((1 - x)/(1 +  x))  "d"x =

• pi/2 - 1

• pi/2 + 1

• pi/2 - 2

• p/2 + 2

MCQ | Q 3

int_1^2 ("e"^(1/x))/(x^2)  "d"x =

• 2sqrt("e")(1 + sqrt("e"))

• sqrt("e")(1 - sqrt("e"))

• sqrt("e")(sqrt("e") - 1)

• sqrt("e")(1 + sqrt("e"))

MCQ | Q 4

int_0^(x/4) sqrt(1 + sin 2x)  "d"x =

• 1/sqrt(2)

• sqrt(2) + 1

• 2sqrt(2)

• 1

MCQ | Q 5

If int_0^1  ("d"x)/(sqrt(1 + x) - sqrt(x)) = "k"/3, then k = ?

• sqrt(2)(2sqrt(2) - 2)

• sqrt(2)/3(2 - 2sqrt(2))

• (2sqrt(2) - 2)/3

• 4sqrt(2)

MCQ | Q 6

int_(pi/5)^((3pi)/10)  sinx/(sinx + cosx)  "d"x =

• pi/10

• pi/20

• pi/6

• pi/12

MCQ | Q 7

int_0^1 (x^2 - 2)/(x^2 + 1)  "d"x =

• 1 - (3pi)/4

• 2 - (3pi)/4

• 1 + (3pi)/4

• 2 + (3pi)/4

MCQ | Q 8

Let I1 = int_"e"^("e"^2)  1/logx  "d"x and I2 = int_1^2 ("e"^x)/x  "d"x then

• I1 = 1/3 "I"_2

• I1 + I2 = 0

• I1 = 2I2

• I1 = I2

MCQ | Q 9

int_0^4 1/sqrt(4x - x^2)  "d"x =

• 0

• π

MCQ | Q 10

int_0^(pi/2) log(tanx)  "d"x =

• pi/8(log2)

• 0

• - pi/8 (log2)

• pi/2 (log2)

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Definite Integration Very Short Answers

#### 1 Mark

Very Short Answers | Q 1

Evaluate: int_(pi/6)^(pi/3) cosx  "d"x

Very Short Answers | Q 2

Evaluate: int_(- pi/4)^(pi/4) x^3 sin^4x  "d"x

Very Short Answers | Q 3

Evaluate: int_0^1 1/(1 + x^2)  "d"x

Very Short Answers | Q 4

Evaluate: int_0^(pi/4) sec^2 x  "d"x

Very Short Answers | Q 5

Evaluate: int_0^1 |x|  "d"x

Very Short Answers | Q 6

Evaluate: int_0^1 1/sqrt(1 - x^2)  "d"x

Very Short Answers | Q 7

Evaluate: int_1^2 x/(1 + x^2)  "d"x

Very Short Answers | Q 8

Evaluate: int_0^1 "e"^x/sqrt("e"^x - 1)  "d"x

Very Short Answers | Q 9

Evaluate: int_0^(pi/2)  (sin2x)/(1 +  sin^2x)  "d"x

Very Short Answers | Q 10

Evaluate: int_0^1(x + 1)^2  "d"x

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Definite Integration Short Answers I

#### 2 Marks

Short Answers I | Q 1

Evaluate: int_(pi/6)^(pi/3) sin^2 x  "d"x

Short Answers I | Q 2

Evaluate: int_0^(pi/2) sqrt(1 - cos 4x)  "d"x

Short Answers I | Q 3

Evaluate: int_0^(pi/2) cos^3x  "d"x

Short Answers I | Q 4

Evaluate: int_0^pi cos^2 x  "d"x

Short Answers I | Q 5

Evaluate: int_0^(pi/4) (tan^3x)/(1 + cos 2x)  "d"x

Short Answers I | Q 6

Evaluate: int_0^(pi/4)  cosx/(4 - sin^2 x)  "d"x

Short Answers I | Q 7

Evaluate: int_1^3 (cos(logx))/x  "d"x

Short Answers I | Q 8

Evaluate: int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x

Short Answers I | Q 9

Evaluate: int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x)  "d"x

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Definite Integration Short Answers II

#### 3 Marks

Short Answers II | Q 1

Prove that: int_"a"^"b" "f"(x)  "d"x = int_"a"^"c""f"(x)  "d"x + int_"c"^"b"  "f"(x)  "d"x, where a < c < b

Short Answers II | Q 2

Prove that: int_"a"^"b" "f"(x)  "d"x = int_"a"^"b" "f"("a" + "b" - x)  "d"x

Short Answers II | Q 3

Prove that: int_0^"a" "f"(x)  "d"x = int_0^"a" "f"("a" - x)  "d"x. Hence find int_0^(pi/2) sin^2x  "d"x

Short Answers II | Q 4

Evaluate: int_0^(pi/2) (sin^4x)/(sin^4x + cos^4x)  "d"x

Short Answers II | Q 5

Evaluate: int_3^8 (11 - x)^2/(x^2 + (11 - x)^2)  "d"x

Short Answers II | Q 6

Evaluate: int_(-1)^1 |5x - 3|  "d"x

Short Answers II | Q 7

Evaluate: int_(-4)^2 1/(x^2 + 4x + 13)  "d"x

Short Answers II | Q 8

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

Short Answers II | Q 9

Evaluate: int_0^1 x* tan^-1x  "d"x

Short Answers II | Q 10

Evaluate: int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2)  "d"x

Short Answers II | Q 11

Evaluate: int_0^(pi/4) sec^4x  "d"x

Short Answers II | Q 12

Evaluate: int_0^(pi/2) 1/(5 + 4cos x)  "d"x

Short Answers II | Q 13

Evaluate: int_0^(pi/2) cos x/((1 + sinx)(2 + sinx))  "d"x

Short Answers II | Q 14

Evaluate: int_(-1)^1 1/("a"^2"e"^x + "b"^2"e"^(-x))  "d"x

Short Answers II | Q 15

Evaluate: int_0^"a" 1/(x + sqrt("a"^2 - x^2))  "d"x

Short Answers II | Q 16

Evaluate: int_0^3 x^2 (3 - x)^(5/2)  "d"x

Short Answers II | Q 17

Evaluate: int_0^1 "t"^2 sqrt(1 - "t")  "dt"

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 Chapter 2 Definite Integration Long Answers III

#### 4 Marks

Long Answers III | Q 1

Prove that: int_0^(2"a") "f"(x)  "d"x = int_0^"a" "f"(x)  "d"x + int_0^"a" "f"(2"a" - x)  "d"x

Long Answers III | Q 2

Prove that: int_(-"a")^"a" "f"(x)  "d"x {:(= 2 int_0^"a" "f" (x)  "d"x ",",  "If"  "f"(x)  "is even function"),(= 0",", "if"  "f"(x)  "is odd function"):}

Long Answers III | Q 3

Evaluate: int_0^(1/2) 1/((1 - 2x^2) sqrt(1 - x^2))  "d"x

Long Answers III | Q 4

Evaluate: int_0^(pi/4)  (sec^2x)/(3tan^2x + 4tan x + 1)  "d"x

Long Answers III | Q 5

Evaluate: int_(1/sqrt(2))^1  (("e"^(cos^-1x))(sin^-1x))/sqrt(1 - x^2)  "d"x

Long Answers III | Q 6

Evaluate: int_0^1 (log(x + 1))/(x^2 + 1)  "d"x

Long Answers III | Q 7

Evaluate: int_0^pi x*sinx*cos^2x* "d"x

Long Answers III | Q 8

Evaluate: int_0^(pi/2)x  sinx  "d"x

Long Answers III | Q 9

Evaluate: int_(-1)^1 (1 + x^2)/(9 - x^2)  "d"x

Long Answers III | Q 10

Evaluate: int_0^1 (1/(1 + x^2)) sin^-1 ((2x)/(1 + x^2))  "d"x

Long Answers III | Q 11

Evaluate: int_0^(pi/4)  (cos2x)/(1 + cos 2x + sin 2x)  "d"x

Long Answers III | Q 12

Evaluate: int_0^(pi/4) log(1 + tanx)  "d"x

Long Answers III | Q 13

Evaluate: int_0^pi 1/(3 + 2sinx + cosx)  "d"x

## Chapter 2: Definite Integration ## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 2 - Definite Integration

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Arts and Science) Maharashtra State Board 2022 chapter 2 Definite Integration are Definite Integral as Limit of Sum, Fundamental Theorem of Integral Calculus, Methods of Evaluation and Properties of Definite Integral.

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