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NCERT solutions for Mathematics Exemplar Class 12 chapter 7 - Integrals [Latest edition]

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Mathematics Exemplar Class 12 - Shaalaa.com
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Chapter 7: Integrals

Solved ExamplesExercise
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Solved Examples [Pages 146 - 163]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 7 IntegralsSolved Examples [Pages 146 - 163]

Short Answer

Solved Examples | Q 1 | Page 146

Integrate `((2"a")/sqrt(x) - "b"/x^2 + 3"c"root(3)(x^2))` w.r.t. x

Solved Examples | Q 2 | Page 147

Evaluate `int (3"a"x)/("b"^2 + "c"^2x^2) "d"x`

Solved Examples | Q 3 | Page 147

Verify the following using the concept of integration as an antiderivative

`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`

Solved Examples | Q 4 | Page 147

Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1

Solved Examples | Q 5 | Page 148

Evaluate `int "dx"/sqrt((x - alpha)(beta - x)), beta > alpha`

Solved Examples | Q 6 | Page 148

Evaluate `int tan^8 x sec^4 x"d"x`

Solved Examples | Q 7 | Page 149

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

Solved Examples | Q 8 | Page 149

Find `int "dx"/(2sin^2x + 5cos^2x)`

Solved Examples | Q 9 | Page 150

Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums

Solved Examples | Q 10 | Page 151

Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`

Solved Examples | Q 11 | Page 152

Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`

Solved Examples | Q 12 | Page 152

Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`

Solved Examples | Q 13 | Page 153

Find `int x^2tan^-1x"d"x`

Solved Examples | Q 14 | Page 153

Find `int sqrt(10 - 4x + 4x^2)  "d"x`

Long Answer

Solved Examples | Q 15 | Page 154

Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`

Solved Examples | Q 16 | Page 154

Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`

Solved Examples | Q 17 | Page 155

Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`

Solved Examples | Q 18 | Page 156

Find `int_0^1 x(tan^-1x)  "d"x`

Solved Examples | Q 19 | Page 158

Evaluate `int_(-1)^2 "f"(x)  "d"x`, where f(x) = |x + 1| + |x| + |x – 1|

Objective Type Questions from 20 to 30

Solved Examples | Q 20 | Page 158

`int "e"^x (cosx - sinx)"d"x` is equal to ______.

  • `"e"^x cos x + "C"`

  • `"e"^x sin x + "C"`

  • `-"e"^x cos x + "C"`

  • `-"e"^x sin x + "C"`

Solved Examples | Q 21 | Page 159

`int "dx"/(sin^2x cos^2x)` is equal to ______.

  • tanx + cotx + C

  • x + cotx)2 + C

  • tanx – cotx + C

  • (tanx – cotx)2 + C

Solved Examples | Q 22 | Page 159

If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then ______.

  • a = `(-1)/8`, b = `7/8`

  • a = `1/8`, b = `7/8`

  • a = `(-1)/8`, b = `(-7)/8`

  • a = `1/8`, b = `(-7)/8`

Solved Examples | Q 23 | Page 160

`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.

  • `int_"a"^"b" "f"(x - "c")"d"x`

  • `int_"a"^"b" "f"(x + "c")"d"x`

  • `int_"a"^"b" "f"(x)"d"x`

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

Solved Examples | Q 24 | Page 160

If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.

  • `"a"/2`

  • `"a"/2 int_0^"a" "f"(x)"d"x`

  • `int_0^"a" "f"(x)"d"x`

  • `"a" int_0^"a" "f"(x)"d"x`

Solved Examples | Q 25 | Page 161

If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.

  • 3

  • 6

  • 9

  • 1

Solved Examples | Q 26 | Page 161

`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.

  • log 2

  • 2 log 2

  • `1/2 log 2`

  • 4 log 2

Solved Examples | Q 27 | Page 162

If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.

  • `"a" - 1 + "e"/2`

  • `"a" + 1 - "e"/2`

  • `"a" - 1 - "e"/2`

  • `"a" + 1 + "e"/2`

Solved Examples | Q 28 | Page 162

`int_(-2)^2 |x cos pix| "d"x` is equal to ______.

  • `8/pi`

  • `4/pi`

  • `2/pi`

  • `1/pi`

Fill in the blanks 29 to 32

Solved Examples | Q 29 | Page 162

`int (sin^6x)/(cos^8x) "d"x` = ______.

Solved Examples | Q 30 | Page 163

`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.

Solved Examples | Q 31 | Page 163

`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.

Solved Examples | Q 32 | Page 163

`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.

Exercise [Pages 163 - 169]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 7 IntegralsExercise [Pages 163 - 169]

Short Answer

Exercise | Q 1 | Page 163

Verify the following:

`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`

Exercise | Q 2 | Page 163

Verify the following:

`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`

Exercise | Q 3 | Page 163

Evaluate the following:

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

Exercise | Q 4 | Page 163

Evaluate the following:

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

Exercise | Q 5 | Page 164

Evaluate the following:

`int ((1 + cosx))/(x + sinx) "d"x`

Exercise | Q 6 | Page 164

Evaluate the following:

`int ("d"x)/(1 + cos x)`

Exercise | Q 7 | Page 164

Evaluate the following:

`int tan^2x sec^4 x"d"x`

Exercise | Q 8 | Page 164

Evaluate the following:

`int (sinx + cosx)/sqrt(1 + sin 2x) "d"x`

Exercise | Q 9 | Page 164

Evaluate the following:

`int sqrt(1 + sinx)"d"x`

Exercise | Q 10 | Page 164

Evaluate the following:

`int x/(sqrt(x) + 1) "d"x`  (Hint: Put  `sqrt(x)` = z)

Exercise | Q 11 | Page 164

Evaluate the following:

`int sqrt(("a" + x)/("a" - x)) "d"x`

Exercise | Q 12 | Page 164

Evaluate the following:

`int x^(1/2)/(1 + x^(3/4)) "d"x`   (Hint: Put `sqrt(x)` = z4)

Exercise | Q 13 | Page 164

Evaluate the following:

`int sqrt(1 + x^2)/x^4 "d"x`

Exercise | Q 14 | Page 164

Evaluate the following:

`int ("d"x)/sqrt(16 - 9x^2)`

Exercise | Q 15 | Page 164

Evaluate the following:

`int "dt"/sqrt(3"t" - 2"t"^2)`

Exercise | Q 16 | Page 164

Evaluate the following:

`int (3x - 1)/sqrt(x^2 + 9) "d"x`

Exercise | Q 17 | Page 164

Evaluate the following:

`int sqrt(5 - 2x + x^2) "d"x`

Exercise | Q 18 | Page 164

Evaluate the following:

`int x/(x^4 - 1) "d"x`

Exercise | Q 19 | Page 164

Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x2 = t

Exercise | Q 20 | Page 164

Evaluate the following:

`int sqrt(2"a"x - x^2)  "d"x`

Exercise | Q 21 | Page 164

Evaluate the following:

`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`

Exercise | Q 22 | Page 164

Evaluate the following:

`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`

Exercise | Q 23 | Page 164

Evaluate the following:

`int (sin^6x + cos^6x)/(sin^2x cos^2x) "d"x`

Exercise | Q 24 | Page 165

Evaluate the following:

`int sqrt(x)/(sqrt("a"^3 - x^3)) "d"x`

Exercise | Q 25 | Page 165

Evaluate the following:

`int (cosx - cos2x)/(1 - cosx) "d"x`

Exercise | Q 26 | Page 165

Evaluate the following:

`int ("d"x)/(xsqrt(x^4 - 1))`  (Hint: Put x2 = sec θ)

Exercise | Q 27 | Page 165

Evaluate the following as limit of sum:

`int _0^2 (x^2 + 3) "d"x`

Exercise | Q 28 | Page 165

Evaluate the following as limit of sum:

`int_0^2 "e"^x "d"x`

Exercise | Q 29 | Page 165

Evaluate the following:

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

Exercise | Q 30 | Page 165

Evaluate the following:

`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`

Exercise | Q 31 | Page 165

Evaluate the following:

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

Exercise | Q 32 | Page 165

Evaluate the following:

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

Exercise | Q 33 | Page 165

Evaluate the following:

`int_0^pi x sin x cos^2x "d"x`

Exercise | Q 34 | Page 165

Evaluate the following:

`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))`  (Hint: Let x = sin θ)

Long Answer

Exercise | Q 35 | Page 165

Evaluate the following:

`int (x^2"d"x)/(x^4 - x^2 - 12)`

Exercise | Q 36 | Page 165

Evaluate the following:

`int (x^2)/((x^2 + "a"^2)(x^2 + "b"^2)) "d"x`

Exercise | Q 37 | Page 165

Evaluate the following:

`int_"b"^pi  x/(1 + sin x)`

Exercise | Q 38 | Page 165

Evaluate the following:

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

Exercise | Q 39 | Page 166

Evaluate the following:

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

Exercise | Q 40 | Page 166

Evaluate the following:

`int sin^-1 sqrt(x/("a" + x)) "d"x`  (Hint: Put x = a tan2θ)

Exercise | Q 41 | Page 166

Evaluate the following:

`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2)  "d"x`

Exercise | Q 42 | Page 166

Evaluate the following:

`int "e"^(-3x) cos^3x  "d"x`

Exercise | Q 43 | Page 166

Evaluate the following:

`int sqrt(tanx)  "d"x`  (Hint: Put tanx = t2)

Exercise | Q 44 | Page 166

Evaluate the following:

`int_0^(pi/2)  "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)

Exercise | Q 45 | Page 166

Evaluate the following:

`int_0^1 x log(1 + 2x)  "d"x`

Exercise | Q 46 | Page 166

Evaluate the following:

`int_0^pi x log sin x "d"x`

Exercise | Q 47 | Page 166

Evaluate the following:

`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`

Objective Type Questions from 48 to 63

Exercise | Q 48 | Page 166

`int (cos2x - cos 2theta)/(cosx - costheta) "d"x` is equal to ______.

  • 2(sinx + xcosθ) + C

  • 2(sinx – xcosθ) + C

  • 2(sinx + 2xcosθ) + C

  • 2(sinx – 2x cosθ) + C

Exercise | Q 49 | Page 167

`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.

  • `sin("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C"`

  • `"cosec"("b" - "a") log|(sin(x - "a"))/(sin(x - "b"))| + "C"`

  • `"cosec"("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C"`

  • `sin("b" - "a")log|(sin("x" - "a"))/(sin(x - "b"))| + "C"`

Exercise | Q 50 | Page 167

`int tan^-1 sqrt(x)  "d"x` is equal to ______.

  • `(x + 1) tan^-1 sqrt(x) - sqrt(x) + "C"`

  • `x tan^-1 sqrt(x) - sqrt(x) + "C"`

  • `sqrt(x) - x tan^-1 sqrt(x) + "C"`

  • `sqrt(x) - (x + 1) tan^-1 sqrt(x) + "C"`

Exercise | Q 51 | Page 167

`int "e"^x ((1 - x)/(1 + x^2))^2  "d"x` is equal to ______.

  • `"e"^x/(1 + x^2) + "C"`

  • `(-"e"^x)/(1 + x^2) + "C"`

  • `"e"^x/(1 + x^2)^2 + "C"`

  • `(-"e"^x)/(1 + x^2)^2 + "C"`

Exercise | Q 52 | Page 167

`int x^9/(4x^2 + 1)^6  "d"x` is equal to ______.

  • `1/(5x)(4 + 1/x^2)^-5 + "C"`

  • `1/5(4 + 1/x^2)^-5 + "C"`

  • `1/(10x)(1 + 4)^-5 + "C"`

  • `1/10(1/x^2 + 4)^-5 + "C"`

Exercise | Q 53 | Page 168

If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.

  • a = `(-1)/10`, b = `(-2)/5` 

  • a = `1/10`, b = `- 2/5`

  • a = `(-1)/10`, b = `2/5`

  • a = `1/10`, b = `2/5`

Exercise | Q 54 | Page 168

`int x^3/(x + 1)` is equal to ______.

  • `x + x^2/2 + x^3/3 - log|1 - x| + "C"`

  • `x + x^2/2 - x^3/3 - log|1 - x| + "C"`

  • `x - x^2/2 - x^3/3 - log|1 + x| + "C"`

  • `x - x^2/2 + x^3/3 - log|1 + x| + "C"`

Exercise | Q 55 | Page 168

`int (x + sinx)/(1 + cosx) "d"x` is equal to ______.

  • log |1 + cosx| + C

  • log |x + sinx| + C

  • `x - tan  x/2 + "C"`

  • `x.tan  x/2 + "C"`

Exercise | Q 56 | Page 168

If `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`, then ______.

  • a = `1/3`, b = 1

  • a = `(-1)/3`, b = 1

  • a = `(-1)/3`, b = –1

  • a = `1/3`, b = –1

Exercise | Q 57 | Page 169

`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.

  • 1

  • 2

  • 3

  • 4

Exercise | Q 58 | Page 169

`int_0^(pi/2) sqrt(1 - sin2x)  "d"x` is equal to ______.

  • `2sqrt(2)`

  • `2(sqrt(2) + 1)`

  • 2

  • `2(sqrt(2) - 1)`

Fill in the blanks 60 to 63.

Exercise | Q 59 | Page 169

`int_0^(pi/2)  cos x "e"^(sinx)  "d"x` is equal to ______.

Exercise | Q 60 | Page 169

`int (x + 3)/(x + 4)^2 "e"^x  "d"x` = ______.

Exercise | Q 61 | Page 169

If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.

Exercise | Q 62 | Page 169

`int sinx/(3 + 4cos^2x) "d"x` = ______.

Exercise | Q 63 | Page 169

The value of `int_(-pi)^pi sin^3x cos^2x  "d"x` is ______.

Chapter 7: Integrals

Solved ExamplesExercise
Mathematics Exemplar Class 12 - Shaalaa.com

NCERT solutions for Mathematics Exemplar Class 12 chapter 7 - Integrals

NCERT solutions for Mathematics Exemplar Class 12 chapter 7 (Integrals) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 12 chapter 7 Integrals are Definite Integrals Problems, Indefinite Integral Problems, Comparison Between Differentiation and Integration, Geometrical Interpretation of Indefinite Integral, Integrals of Some Particular Functions, Indefinite Integral by Inspection, Properties of Indefinite Integral, Integration Using Trigonometric Identities, Introduction of Integrals, Evaluation of Definite Integrals by Substitution, Properties of Definite Integrals, Fundamental Theorem of Calculus, Definite Integral as the Limit of a Sum, Evaluation of Simple Integrals of the Following Types and Problems, Methods of Integration: Integration by Parts, Methods of Integration: Integration Using Partial Fractions, Methods of Integration: Integration by Substitution, Integration as an Inverse Process of Differentiation.

Using NCERT Class 12 solutions Integrals exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

Get the free view of chapter 7 Integrals Class 12 extra questions for Mathematics Exemplar Class 12 and can use Shaalaa.com to keep it handy for your exam preparation

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