#### Chapters

Chapter 2: Basic Algebra

Chapter 3: Trigonometry

Chapter 4: Combinatorics and Mathematical Induction

Chapter 5: Binomial Theorem, Sequences and Series

Chapter 6: Two Dimensional Analytical Geometry

Chapter 7: Matrices and Determinants

Chapter 8: Vector Algebra

Chapter 9: Differential Calculus - Limits and Continuity

Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation

Chapter 11: Integral Calculus

Chapter 12: Introduction to probability theory

## Chapter 11: Integral Calculus

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.1 [Page 188]

Integrate the following with respect to x:

x^{11}

Integrate the following with respect to x:

`1/x^7`

Integrate the following with respect to x:

`root(3)(x^4)`

Integrate the following with respect to x:

`(x^5)^(1/8)`

Integrate the following with respect to x:

`1/(sin^2x)`

Integrate the following with respect to x:

`tanx/cosx`

Integrate the following with respect to x:

`cosx/(sin^2x)`

Integrate the following with respect to x:

`1/(cos^2x)`

Integrate the following with respect to x:

12^{3}

Integrate the following with respect to x:

`(x^24)/(x^25)`

Integrate the following with respect to x:

e^{x}

Integrate the following with respect to x:

(1 + x^{2})^{–1 }

Integrate the following with respect to x:

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.2 [Page 190]

Integrate the following functions with respect to x:

(x + 5)^{6}

Integrate the following functions with respect to x:

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

Integrate the following functions with respect to x:

`sqrt(3x + 2)`

Integrate the following functions with respect to x:

sin 3x

Integrate the following functions with respect to x:

cos(5 – 11x)

Integrate the following functions with respect to x:

cosec^{2} (5x – 7)

Integrate the following functions with respect to x:

`"e"^(3x - 6)`

Integrate the following functions with respect to x:

`"e"^(8 - 7x)`

Integrate the following functions with respect to x:

`1/(6 - 4x)`

Integrate the following functions with respect to x:

`sec^2 x/5`

Integrate the following functions with respect to x:

cosec(5x + 3) cot(5x + 3)

Integrate the following functions with respect to x:

30 sec(2 – 15x) tan(2 – 15x)

Integrate the following functions with respect to x:

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

Integrate the following functions with respect to x:

`1/sqrt(1 - 81x^2)`

Integrate the following functions with respect to x:

`1/sqrt(1 + 36x^2)`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.3, Exercise 11. [Page 192]

Integrate the following with respect to x:

`(x + 4)^5 + 5/(2 - 5x)^4 - "cosec"^2 (3x - 1)`

Integrate the following with respect to x:

`4 cos(5 - 2x) + 9"e"^(3x - 6) + 24/(6 - 4x)`

Integrate the following with respect to x:

`sec^2 x/5 + 18 cos2x + 10sec(5x + 3) tan(5x + 3)`

Integrate the following with respect to x:

`8/sqrt(1 - (4x)^2) + 27/sqrt(1 - 9x^2) - 15/(1 + 25x^2)`

Integrate the following with respect to x:

`6/(1 + (3x + 2)^2) - 12/sqrt(1 - (3 - 4x)^2`

Integrate the following with respect to x:

`1/3 cos(x/3 - 4) + 7/(7x + 9) + "e"^(x/5 + 3)`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.4 [Pages 195 - 196]

If f'(x) = 4x – 5 and f(2) = 1, find f(x)

If f'(x) = 9x^{2} – 6x and f(0) = – 3 find f(x)

If f” (x) = 12x – 6 and f(1) = 30, f’(1) = 5, find f(x)

A ball is thrown vertically upward from the ground with an initial velocity of 39.2 m/sec. If the only force considered is that attributed to the acceleration due to gravity, find how long will it take for the ball to strike the ground?

A ball is thrown vertically upward from the ground with an initial velocity of 39.2 m/sec. If the only force considered is that attributed to the acceleration due to gravity, find the speed with which will it strike the ground?

A ball is thrown vertically upward from the ground with an initial velocity of 39.2 m/sec. If the only force considered is that attributed to the acceleration due to gravity, find how high the ball will rise?

A wound is healing in such a way that t days since Sunday the area of the wound has been decreasing at a rate of `- 6/("t" + 2)^2` cm^{2} per day where 0 < t ≤ 8. If on Monday the area of the wound was 1.4 cm^{2}. What was the area of the wound on Sunday?

A wound is healing in such a way that t days since Sunday the area of the wound has been decreasing at a rate of `-6/("t" + 2)^2` cm^{2} per day where 0 < t ≤ 8. If on Monday the area of the wound was 1.4 cm^{2}. What is the anticipated area of the wound on Thursday if it continues to heal at the same rate?

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.5 [Page 202]

Integrate the following functions with respect to x :

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

Integrate the following functions with respect to x :

`[sqrt(x) + 1/sqrt(x)]^2`

Integrate the following functions with respect to x :

(2x – 5)(3x + 4x)

Integrate the following functions with respect to x :

cot^{2}x + tan^{2}x

Integrate the following functions with respect to x :

`(cos2x - cos 2 alpha)/(cosx - cos alpha)`

Integrate the following functions with respect to x :

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

Integrate the following functions with respect to x :

`(3 + 4cosx)/(sin^2x)`

Integrate the following functions with respect to x :

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

Integrate the following functions with respect to x :

`(sin4x)/sinx`

Integrate the following functions with respect to x :

cos 3x cos 2x

Integrate the following functions with respect to x :

sin^{2} 5x

Integrate the following functions with respect to x :

`(1 + cos 4x)/(cos x - tan x)`

Integrate the following functions with respect to x :

`"e"^(x log "a") "e"^x`

Integrate the following functions with respect to x :

`(3x + 4) sqrt(3x + 7)`

Integrate the following functions with respect to x :

`(8^(1 + x) + 4^(1 - x))/2^x`

Integrate the following functions with respect to x :

`1/(sqrt(x + 3) - sqrt(x - 4))`

Integrate the following functions with respect to x :

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

Integrate the following functions with respect to x :

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

Integrate the following functions with respect to x :

`(3x - 9)/((x - 1)(x + 2)(x^2 + 1))`

Integrate the following functions with respect to x :

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.6 [Page 206]

Integrate the following with respect to x :

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

Integrate the following with respect to x :

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

Integrate the following with respect to x :

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

Integrate the following with respect to x :

`(10x^9 + 10^x log_"e" 10)/(10^x + x^10)`

Integrate the following with respect to x :

`(sin sqrt(x))/sqrt(x)`

Integrate the following with respect to x :

`cot x/(log(sin x))`

Integrate the following with respect to x :

`("cosec" x)/(log(tan x/2))`

Integrate the following with respect to x :

`(sin 2x)/("a"^2 + "b"^2 sin^2x)`

Integrate the following with respect to x :

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

Integrate the following with respect to x :

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

Integrate the following with respect to x :

`1/(x log x log (log x))`

Integrate the following with respect to x :

`alpha beta x^(alpha - 1) "e"^(- beta x^alpha)`

Integrate the following with respect to x :

`tan x sqrt(sec x)`

Integrate the following with respect to x :

x(1 – x)^{17}

Integrate the following with respect to x :

sin^{5}x cos^{3}x

Integrate the following with respect to x :

`cosx/(cos(x - "a"))`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.7 [Page 210]

Integrate the following with respect to x:

9xe^{3x}

Integrate the following with respect to x:

x sin 3x

Integrate the following with respect to x:

25xe^{–5x}

Integrate the following with respect to x:

x sec x tan x

Integrate the following with respect to x:

x log x

Integrate the following with respect to x:

27x^{2}e^{3x}

Integrate the following with respect to x:

x^{2} cos x

Integrate the following with respect to x:

x^{3} sin x

Integrate the following with respect to x:

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

Integrate the following with respect to x:

x^{5}e^{x2}

Integrate the following with respect to x:

`tan^-1 ((8x)/(1 - 16x^2))`

Integrate the following with respect to x:

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.8 [Page 212]

Integrate the following with respect to x:

`"e"^("a"x) cos"b"x`

Integrate the following with respect to x:

`"e"^(2x) sinx`

Integrate the following with respect to x:

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

Integrate the following with respect to x:

`"e"^(- 3x) sin 2x`

Integrate the following with respect to x:

`"e"^(- 4x) sin 2x`

Integrate the following with respect to x:

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.9 [Page 213]

Integrate the following with respect to x:

`"e"^x (tan x + log sec x)`

Integrate the following with respect to x:

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

Integrate the following with respect to x:

`"e"^x sec x(1 + tan x)`

Integrate the following with respect to x:

`"e"^x ((2 + sin 2x)/(1 + cos 2x))`

Integrate the following with respect to x:

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

Integrate the following with respect to x:\

`logx/(1 + log)^2`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.10 [Page 219]

Find the integrals of the following:

`1/(4 - x^2)`

Find the integrals of the following:

`1/(25 - 4x^2)`

Find the integrals of the following:

`1/(9x^2 - 4)`

Find the integrals of the following:

`1/(6x - 7 - x^2)`

Find the integrals of the following:

`1/((x + 1)^2 - 25)`

Find the integrals of the following:

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

Find the integrals of the following:

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

Find the integrals of the following:

`1/sqrt(x^2 - 4x + 5)`

Find the integrals of the following:

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.11 [Page 222]

Integrate the following with respect to x:

`(2x - 3)/(x^2 + 4x - 12)`

Integrate the following with respect to x:

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

Integrate the following with respect to x:

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

Integrate the following with respect to x:

`(2x + 1)/sqrt(9 + 4x - x^2)`

Integrate the following with respect to x:

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

Integrate the following with respect to x:

`(2x + 3)/sqrt(x^2 + 4x + 1)`

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.12 [Page 225]

Integrate the following functions with respect to x:

`sqrt(x^2 + 2x + 10)`

Integrate the following functions with respect to x:

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

Integrate the following functions with respect to x:

`sqrt((6 - x)(x - 4))`

Integrate the following functions with respect to x:

`sqrt(9 - (2x + 5)^2`

Integrate the following functions with respect to x:

`sqrt(81 + (2x + 1)^2`

Integrate the following functions with respect to x:

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide Chapter 11 Integral CalculusExercise 11.13 [Pages 225 - 227]

Choose the correct alternative:

If `int f(x) "d"x = g(x) + "c",` then `int f(x)g"'"(x) "d"x`

`int (f(x))^2 "d"x`

`int f(x) g(x) "d"x`

`int f"'"(x)g(x) "d"x`

`int (g(x))^2 "d"x`

Choose the correct alternative:

If `int 3^(1/x)/x^2 "d"x = "k"(3^(1/x)) + "c"`, then the value of k is

log 3

– log 3

`- 1/log 3`

`1/log 3`

Choose the correct alternative:

If `int f"'"(x)"e"^(x^2) "d"x = (x - 1)"e"^(x^2) + "c"`, then`f(x)` is

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

`x^3/2 + 3x^2 + 4x + "c"`

`x^3 + 4x^2 + 6x + "c"`

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

Choose the correct alternative:

The gradient (slope) of a curve at any point (x, y) is `(x^2 - 4)/x^2`. If the curve passes through the point (2, 7), then the equation of the curve is

y = `x + 4/x + 3`

y = `x + 4/x + 4`

y = x

^{2}+ 3x + 4y = x

^{2}– 3x + 6

Choose the correct alternative:

`int ("e"^x (1 + x))/(cos^2(x"e"^x)) "d"x` is

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

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

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

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

Choose the correct alternative:

`int sqrt(tanx)/(sin2x) "d"x` is

`sqrt(tan x) + "c"`

`2sqrt(tan x) + "c"`

`1/2 sqrt(tan x) + "c"`

`1/4 sqrt(tan x) + "c"`

Choose the correct alternative:

`int sin^2x "d"x` is

`(- 3)/4 cos x - (cos3x)/12 + "c"`

`3/4 cos x + (cos3x)/12 + "c"`

`(- 3)/4 cos x + (cos3x)/12 + "c"`

`(- 3)/4 sin x - (sin3x)/12 + "c"`

Choose the correct alternative:

`int ("e"^(6 log x) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x` is

x + c

`x^3/3 + "c"`

`3/x^3 + "c"`

`1/x^2 + "c"`

Choose the correct alternative:

`int secx/sqrt(cos2x) "d"x` is

`tan^-1 (sin x) + "c"`

`2sin^-1 (tan x) + "c"`

`tan^-1 (cos x) + "c"`

`sin^-1 (tan x) + "c"`

Choose the correct alternative:

`int tan^-1 sqrt((1 - cos 2x)/(1 + cos 2x)) "d"x` is

x

^{2}+ c2x

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

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

Choose the correct alternative:

`int 2^(3x + 5) "d"x` is

`(3(2^(3x + 5)))/log2 + "c"`

`2^(3x + 5)/(2log(3x + 5)) + "c"`

`2^(3x + 5)/(2log3) + "c"`

`2^(3x + 5)/(3log2) + "c"`

Choose the correct alternative:

`int (sin^2x - cos^8x)/(1 - 2sin^x cosx) "d"x` is

`1/2 sin 2x + "c"`

`- 1/2 sin 2x + "c"`

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

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

Choose the correct alternative:

`int ("e"^x(x^2 tan^-1x + tan^-1x + 1))/(x^2 + 1) "d"x` is

e

^{x}tan^{-1}(x + 1) + ctan

^{-1}(e^{x}) + c`"e"^x (tan^-1 x)^2/2 + "c"`

e

^{x}tan^{-1}x + c

Choose the correct alternative:

`int (x^2 + cos^2x)/(x^2 + 1) "cosec"^2 x/("d"x)` is

cot x + sin

^{-1}x + c– cot x + tan

^{-1}x + c– tan x + cot

^{-1}x + c– cot x – tan

^{-1}x + c

Choose the correct alternative:

`int x^2 cos x "d"x` is

x

^{2}sin x + 2x cos x – 2 sin x + cx

^{2}sin x – 2x cos x – 2 sin x + c– x

^{2}sin x + 2x cos x + 2 sin x + c– x

^{2}sin x – 2x cos x + 2 sin x + c

Choose the correct alternative:

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

`sqrt(1 - x^2) + sin^-1x + "c"`

`sin^-1x - sqrt(1 - x^2) + "c"`

`log |x + sqrt(1 - x^2)| - sqrt(1 - x^2) + "c"`

`sqrt(1 - x^2) + log|x + sqrt(1 - x^2)| + "c"`

Choose the correct alternative:

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

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

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

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

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

Choose the correct alternative:

`int "e"^(- 4x) cos "d"x` is

`"e"^(- 4x)/17 [4cos x - sin x] + "c"`

`"e"^(- 4x)/17 [- 4cos x + sin x] + "c"`

`"e"^(- 4x)/17 [4cos x + sin x] + "c"`

`"e"^(- 4x)/17 [- 4cos x - sin x] + "c"`

Choose the correct alternative:

`int (sec^2x)/(tan^2 x - 1) "d"x`

`2 log |(1 - tan x)/(1 + tan x)| + "c"`

`log |(1 + tan x)/(1 - tan x)| + "c"`

`1/2 log |(tan x + 1)/(tan x - 1)| + "c"`

`1/2 log |(tan x - 1)/(tan x + 1)| + "c"`

Choose the correct alternative:

`int "e"^(- 7x) sin 5x "d"x` is

`"e"^(- 7x)/74 [- 7 sin 5x - 5 cos 5x] + "c"`

`"e"^(- 7x)/74 [7 sin 5x + 5 cos 5x] + "c"`

`"e"^(- 7x)/74 [7 sin 5x - 5 cos 5x] + "c"`

`"e"^(- 7x)/74 [- 7 sin 5x + 5 cos 5x] + "c"`

Choose the correct alternative:

`int x^2 "e"^(x/2) "d"x` is

`x^2 "e"^(x/2) - 4x"e"^(x/2) - 8"e"^(x/2) + "c"`

`2x^2 "e"^(x/2) - 8x"e"^(x/2) - 16"e"^(x/2) + "c"`

`2x^2 "e"^(x/2) - 8x"e"^(x/2) + 16"e"^(x/2) + "c"`

`x^2 "e"^(x/2)/2 - (x"e"^(x/2))/4 + "e"^(x/2)/8 + "c"`

Choose the correct alternative:

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

`sqrt(x^2 - 1) - 2log |x + sqrt(x^2 - 1)| + "c"`

`sin^-1x - 2log |x + sqrt(x^2 - 1)| + "c"`

`2log |x + sqrt(x^2 - 1)| - sin^-1x + "c"`

`sqrt(x^2 - 1) + 2log |x + sqrt(x^2 - 1)| + "c"`

Choose the correct alternative:

`int 1/(x sqrt(log x)^2 - 5) "d"x` is

`log |x + sqrt(x^2 - 5)| + "c"`

`log|log x + sqrt(log x - 5)| + "c"`

`log|log x + sqrt((log x)^2 - 5)| + "c"`

`log|log x - sqrt((log x)^2 - 5)| + "c"`

Choose the correct alternative:

`int sin sqrt(x) "d"x` is

`2(- sqrt(x) cos sqrt(x) + sin sqrt(x)) + "c"`

`2(- sqrt(x) cos sqrt(x) - sin sqrt(x)) + "c"`

`2(- sqrt(x) sin sqrt(x) - cos sqrt(x)) + "c"`

`2(- sqrt(x) sin sqrt(x) + cos sqrt(x)) + "c"`

Choose the correct alternative:

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

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

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

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

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

## Chapter 11: Integral Calculus

## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 11 - Integral Calculus

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Concepts covered in Class 11th Mathematics Volume 1 and 2 Answers Guide chapter 11 Integral Calculus are Integral Calculus, Newton-leibnitz Integral, Basic Rules of Integration, Integrals of the Form, Properties of Integrals, Simple Applications, Methods of Integration.

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