#### Chapters

Chapter 2: Integral Calculus – 1

Chapter 3: Integral Calculus – 2

Chapter 4: Differential Equations

Chapter 5: Numerical Methods

Chapter 6: Random Variable and Mathematical expectation

Chapter 7: Probability Distributions

Chapter 8: Sampling techniques and Statistical Inference

Chapter 9: Applied Statistics

Chapter 10: Operations Research

## Chapter 2: Integral Calculus – 1

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.1 [Page 30]

Integrate the following with respect to x.

`sqrt(3x + 5)`

Integrate the following with respect to x.

`(9x^2 - 4/x^2)^2`

Integrate the following with respect to x.

(3 + x)(2 – 5x)

Integrate the following with respect to x.

`sqrt(x)(x^3 - 2x + 3)`

Integrate the following with respect to x.

`(8x + 13)/sqrt(4x + 7)`

Integrate the following with respect to x.

`1/(sqrt(x + 1) + sqrt(x - 1))`

Integrate the following with respect to x.

If f'(x) = x + b, f(1) = 5 and f(2) = 13, then find f(x)

Integrate the following with respect to x.

If f'(x) = 8x^{3} – 2x and f(2) = 8, then find f(x)

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.2 [Page 31]

Integrate the following with respect to x.

`(sqrt(2x) - 1/sqrt(2x))^2`

Integrate the following with respect to x.

`(x^4 - x^2 + 2)/(x - 1)`

Integrate the following with respect to x.

`x^3/(x + 2)`

Integrate the following with respect to x.

`(x^3 + 3x^2 - 7x + 11)/(x + 5)`

Integrate the following with respect to x.

`(3x + 2)/((x - 2)(x - 3))`

Integrate the following with respect to x.

`(4x^2 + 2x + 6)/((x + 1)^2(x - 3))`

Integrate the following with respect to x.

`(3x^2 - 2x + 5)/((x - 1)(x^2 + 5))`

Integrate the following with respect to x.

If f'(x) = `1/x` and f(1) = `pi/4`, then find f(x)

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.3 [Page 32]

Integrate the following with respect to x.

`"e"^(xlog"a") + "e"^("a"log"a") - "e"^("n"logx)`

Integrate the following with respect to x.

`("a"^x - "e"^(xlog"b"))/("e"^(x log "a") "b"^x)`

Integrate the following with respect to x.

`("e"^x + 1)^2 "e"^x`

Integrate the following with respect to x.

`("e"^(3x) - "e"^(-3x))/"e"^x`

Integrate the following with respect to x.

`("e"^(3x) +"e"^(5x))/("e"^x + "e"^-x)`

Integrate the following with respect to x.

`[1 - 1/2]"e"^((x + 1/x))`

Integrate the following with respect to x.

`1/(x(log x)^2`

Integrate the following with respect to x.

If f'(x) = e^{x} and f(0) = 2, then find f(x)

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.4 [Page 33]

Integrate the following with respect to x.

2 cos x – 3 sin x + 4 sec^{2}x – 5 cosec^{2}x

Integrate the following with respect to x.

sin^{3}x

Integrate the following with respect to x.

`(cos 2x + 2sin^2x)/(cos^2x)`

Integrate the following with respect to x.

`1/(sin^2x cos^2x) ["Hint:" sin^2x + cos^2x = 1]`

Integrate the following with respect to x.

`sqrt(1 - sin 2x)`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.5 [Page 35]

Integrate the following with respect to x.

xe^{–x}

Integrate the following with respect to x.

x^{3}e^{3x}

Integrate the following with respect to x.

log x

Integrate the following with respect to x.

x log x

Integrate the following with respect to x.

x^{n} log x

Integrate the following with respect to x.

`x^5 "e"^x`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.6 [Page 38]

Integrate the following with respect to x.

`(2x + 5)/(x^2 + 5x - 7)`

Integrate the following with respect to x.

`("e"^(3logx))/(x^4 + 1)`

Integrate the following with respect to x.

`"e"^(2x)/("e"^(2x) - 2)`

Integrate the following with respect to x.

`(log x)^3/x`

Integrate the following with respect to x.

`(6x + 7)/sqrt(3x^2 + 7x - 1)`

Integrate the following with respect to x.

`(4x + 2) sqrt(x^2 + x + 1)`

Integrate the following with respect to x.

x^{8}(1 + x^{9})^{5}

Integrate the following with respect to x.

`(x^("e" - 1) + "e"^(x - 1))/(x^"e" + "e"^x)`

Integrate the following with respect to x

`1/(x log x)`

Integrate the following with respect to x.

`x/(2x^4 - 3x^2 - 2)`

Integrate the following with respect to x.

e^{x}(1 + x) log(xe^{x})

Integrate the following with respect to x.

`1/(x^2(x^2 + 1))`

Integrate the following with respect to x.

`"e"^x [1/x^2 - 2/x^3]`

Integrate the following with respect to x.

`"e"^x [(x - 1)/(x + 1)^3]`

Integrate the following with respect to x.

`"e"^(3x) [(3x - 1)/(9x^2)]`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.7 [Page 42]

Integrate the following with respect to x.

`1/(9 - 16x^2)`

Integrate the following with respect to x.

`1/(9 - 8x - x^2)`

Integrate the following with respect to x.

`1/(2x^2 - 9)`

Integrate the following with respect to x.

`1/(x^2 - x - 2)`

Integrate the following with respect to x.

`1/(x^2 + 3x + 2)`

Integrate the following with respect to x.

`1/(2x^2 + 6x - 8)`

Integrate the following with respect to x.

`"e"^x/("e"^(2x) - 9)`

Integrate the following with respect to x.

`1/sqrt(9x^2 - 7)`

Integrate the following with respect to x.

`1/sqrt(x^2 + 6x + 13)`

Integrate the following with respect to x.

`1/sqrt(x^2 - 3x + 2)`

Integrate the following with respect to x.

`x^3/sqrt(x^8 - 1)`

Integrate the following with respect to x.

`sqrt(1 + x + x^2)`

Integrate the following with respect to x.

`sqrt(x^2 - 2)`

Integrate the following with respect to x.

`sqrt(4x^2 - 5)`

Integrate the following with respect to x.

`sqrt(2x^2 + 4x + 1)`

Integrate the following with respect to x.

`1/(x + sqrt(x^2 - 1)`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.8 [Page 47]

Using second fundamental theorem, evaluate the following:

`int_0^1 "e"^(2x) "d"x`

Using second fundamental theorem, evaluate the following:

`int_0^(1/4) sqrt(1 - 4) "d"x`

Using second fundamental theorem, evaluate the following:

`int_1^2 (x "d"x)/(x^2 + 1)`

Using second fundamental theorem, evaluate the following:

`int_0^3 ("e"^x "d"x)/(1 + "e"^x)`

Using second fundamental theorem, evaluate the following:

`int_0^1 x"e"^(x^2) "d"x`

Using second fundamental theorem, evaluate the following:

`int_1^"e" ("d"x)/(x(1 + logx)^3`

Using second fundamental theorem, evaluate the following:

`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7) "d"x`

Using second fundamental theorem, evaluate the following:

`int_0^(pi/2) sqrt(1 + cos x) "d"x`

Using second fundamental theorem, evaluate the following:

`int_1^2 (x - 1)/x^2 "d"x`

Evaluate the following:

`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`

Evaluate the following:

`int_0^2 "f"(x) "d"x` where f(x) = `{{:(3 - 2x - x^2",", x ≤ 1),(x^2 + 2x - 3",", 1 < x ≤ 2):}`

Evaluate the following:

`int_(-1)^1 "f"(x) "d"x` where f(x) = `{{:(x",", x ≥ 0),(-x",", x < 0):}`

Evaluate the following:

f(x) = `{{:("c"x",", 0 < x < 1),(0",", "otherwise"):}` Find 'c" if `int_0^1 "f"(x) "d"x` = 2

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.9 [Page 50]

Evaluate the following using properties of definite integral:

`int_(- pi/4)^(pi/4) x^3 cos^3 x "d"x`

Evaluate the following using properties of definite integral:

`int_(- pi/2)^(pi/2) sin^2theta "d"theta`

Evaluate the following using properties of definite integral:

`int_(-1)^1 log ((2 - x)/(2 + x)) "d"x`

Evaluate the following using properties of definite integral:

`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`

Evaluate the following using properties of definite integral:

`int_0^1 log (1/x - 1) "d"x`

Evaluate the following using properties of definite integral:

`int_0^1 x/((1 - x)^(3/4)) "d"x`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.10 [Page 51]

Evaluate the following:

Γ(4)

Evaluate the following:

`Γ (9/2)`

Evaluate the following:

`int_0^oo "e"^(-mx) x^6 "d"x`

Evaluate the following:

`int_0^oo "e"^(-4x) x^4 "d"x`

Evaluate the following:

`int_0^oo "e"^(- x/2) x^5 "d"x`

If f(x) = `{{:(x^2"e"^(-2x)",", x ≥ 0),(0",", "otherwise"):}`, then evaluate `int_0^oo "f"(x) "d"x`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.11 [Page 53]

Evaluate the following integrals as the limit of the sum:

`int_0^1 (x + 4) "d"x`

Evaluate the following integrals as the limit of the sum:

`int_1^3 x "d"x`

Evaluate the following integrals as the limit of the sum:

`int_1^3 (2x + 3) "d"x`

Evaluate the following integrals as the limit of the sum:

`int_0^1 x^2 "d"x`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Exercise 2.12 [Pages 53 - 55]

#### MCQ

Choose the correct alternative:

`int 1/x^3 "d"x` is

`(-3)/(x^2) + "c"`

`(-1)/(2x^2) + "c"`

`(-1)/(3x^2) + "c"`

`(-2)/(x^2) + "c"`

Choose the correct alternative:

`int 2^x "d"x` is

2

^{x}log 2 + c2

^{x}+ c`2^x/log2 + "c"`

`log2/2^x + "c"`

Choose the correct alternative:

`int (sin2x)/(2sinx) "d"x` is

sin x + c

`1/2 sin x + "c"`

`cos x + "c"`

`1/2 cos x + "c"`

Choose the correct alternative:

`int (sin5x - sinx)/(cos3x) "d"x` is

− cos 2x + c

− cos 2x + c

`- 1/4 cos 2x + "c"`

− 4 cos 2x + c

Choose the correct alternative:

`int logx/x "d"x, x > 0` is

`1/2 (log x)^2 + "c"`

`- 1/2 (log x)^2`

`2/x^2 + "c"`

`2/x^2 - "c"`

Choose the correct alternative:

`int "e"^x/sqrt(1 + "e"^x) "d"x` is

`"e"^x/sqrt(1 + "e"^x) + "c"`

`2sqrt(1 + "e"^x) + "c"`

`sqrt(1 + "e"^x) + "c"`

`"e"^x sqrt(1 + "e"^x) + "c"`

Choose the correct alternative:

`int sqrt("e"^x) "d"x` is

`sqrt("e"^x) + "c"`

`2sqrt("e"^x) + "c"`

`1/2 sqrt("e"^x) + "c"`

`1/(2sqrt("e"^x)) + "c"`

Choose the correct alternative:

`int "e"^(2x) [2x^2 + 2x] "d"x`

`"e"^(2x)x^2 + "c"`

`x"e"^(2x) + "c"`

`2x^2"e"^2 + "c"`

`(x^2"e"^x)/2 + "c"`

Choose the correct alternative:

`int "e"^x/("e"^x + 1) "d"x` is

`log|"e"^x/("e"^x + 1)| + "c"`

`log|("e"^x + 1)/"e"| + "c"`

`log|"e"^x| + "c"`

`log|"e"^x + 1| + "c"`

Choose the correct alternative:

`int[9/(x - 3) - 1/(x + 1)] "d"x` is

`log |x - 3| - log |x + 1| + "c"`

`log |x - 3| + log |x + 1| + "c"`

`9log |x - 3| - log |x + 1| + "c"`

`9log |x - 3| + log |x + 1| + "c"`

Choose the correct alternative:

`int (2x^3)/(4 + x^4) "d"x` is

`log |4 + x^4| + "c"`

`1/2 log |4 + x^4| + "c"`

`1/4 log |4 + x^4| + "c"`

`log |2x^3/(4 + x^4) + "c"`

Choose the correct alternative:

`int ("d"x)/sqrt(x^2 - 36) + "c"`

`sqrt(x^2 - 36) + "c"`

`log |x + sqrt(x^2 - 36)| + "c"`

`log |x - sqrt(x^2 - 36)| + "c"`

`log |x^2 + sqrt(x^2 - 36)| + "c"`

Choose the correct alternative:

`int (2x + 3)/sqrt(x^2 + 3x + 2) "d"x` is

`sqrt(x^2 + 3x + 2) + "c"`

`2sqrt(x^2 + 3x + 2) + "c"`

`log(x^2 + 3x + 2) + "c"`

`2/3(x^2 + 3x + 2) + "c"`

Choose the correct alternative:

`int_0^1 (2x + 1) "d"x` is

1

2

3

4

Choose the correct alternative:

`int_2^4 ("d"x)/x` is

log 4

0

log 2

log 8

Choose the correct alternative:

`int_0^oo "e"^(-2x) "d"x` is

0

1

2

`1/2`

Choose the correct alternative:

`int_(-1)^1 x^3 "e"^(x^4) "d"x` is

1

`2 int_0^1 x^3 "e"^(x^4) "d"x`

0

`"e"^(x^4)`

Choose the correct alternative:

If f(x) is a continuous function and a < c < b, then `int_"a"^"c" f(x) "d"x + int_"c"^"b" f(x) "d"x` is

`int_"a"^"b" f(x) "d"x - int_"a"^"c" f(x) "d"x`

`int_"a"^"c" f(x) "d"x - int_"a"^"b" f(x) "d"x`

`int_"a"^"b" f(x) "d"x`

0

Choose the correct alternative:

The value of `int_(- pi/2)^(pi/2) cos x "d"x` is

0

2

1

4

Choose the correct alternative:

`int_0^1 sqrt(x^4 (1 - x)^2) "d"x` is

`1/12`

`(-7)/12`

`7/12`

`(-1)/12`

Choose the correct alternative:

If `int_0^1 f(x) "d"x = 1, int_0^1 x f(x) "d"x = "a"`, and `int_0^1 x^2 f(x) "d"x = "a"^2`, then `int_0^1 ("a" - x)^2 f(x) "d"x` is

4a

^{2}0

2a

^{2}1

Choose the correct alternative:

The value of `int_2^3 f(5 - 3) "d"x - int_2^3 f(x) "d"x` is

1

0

– 1

5

Choose the correct alternative:

`int_0^4 (sqrt(x) + 1/sqrt(x)) "d"x` is

`20/3`

`21/3`

`28/3`

`1/3`

Choose the correct alternative:

`int_0^(pi/3) tan x "d"x` is

log 2

0

`log sqrt(2)`

2 log 2

Choose the correct alternative:

Using the factorial representation of the gamma function, which of the following is the solution for the gamma function Γ(n) when n = 8 is

5040

5400

4500

5540

Choose the correct alternative:

Γ(n) is

(n – 1)!

n!

n Γ(n)

(n – 1) Γ(n)

Choose the correct alternative:

Γ(1) is

0

1

n

n!

Choose the correct alternative:

If n > 0, then Γ(n) is

`int_0^1 "e"^-x x^("n" - 1) "d"x`

`int_0^1 "e"^-x x^"n" "d"x`

`int_0^oo "e"^x x^-"n" "d"x`

`int_0^oo "e"^-x x^("n" - 1) "d"x`

Choose the correct alternative:

`Γ(3/2)`

`sqrt(pi)`

`sqrt(pi)/2`

`2sqrt(pi)`

`3/2`

Choose the correct alternative:

`int_0^oo x^4"e"^-x "d"x` is

12

4

4!

64

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 2 Integral Calculus – 1Miscellaneous problems [Page 55]

Evaluate the following integral:

`int 1/(sqrt(x + 2) - sqrt(x + 3)) "d"x`

Evaluate the following integral:

`int ("d"x)/(2 - 3x - 2x^2)`

Evaluate the following integral:

`int ("d"x)/("e"^x + 6 + 5"e"^-x)`

Evaluate the following integral:

`int sqrt(2x^2 - 3) "d"x`

Evaluate the following integral:

`sqrt(9x^2 + 12x + 3) "d"x`

Evaluate the following integral:

`int (x + 1)^2 log x "d"x`

Evaluate the following integral:

`int log (x - sqrt(x^2 - 1)) "d"x`

Evaluate the following integral:

`int_0^1 sqrt(x(x - 1)) "d"x`

Evaluate the following integral:

`int_(-1)^1 x^2 "e"^(-2x) "d"x`

Evaluate the following integral:

`int_0^3 (x dx)/(sqrt(x + 1)+ sqrt(5x + 1))`

## Chapter 2: Integral Calculus – 1

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 2 - Integral Calculus – 1

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Concepts covered in Class 12th Business Mathematics and Statistics Answers Guide chapter 2 Integral Calculus – 1 are Indefinite Integrals, Definite Integrals.

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