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

## Chapter 7: Integrals

Solved ExamplesExercise
Solved Examples [Pages 146 - 163]

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

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

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]

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)

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

## 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.

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