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Chapters
2: Inverse Trigonometric Functions
3: Matrices
4: Determinants
5: Continuity And Differentiability
6: Application Of Derivatives
▶ 7: Integrals
8: Application Of Integrals
9: Differential Equations
10: Vector Algebra
11: Three Dimensional Geometry
12: Linear Programming
13: Probability
![NCERT Exemplar solutions for Mathematics Exemplar [English] Class 12 chapter 7 - Integrals - Shaalaa.com](/images/mathematics-exemplar-english-class-12_6:fb5456d270e742c88681012a2276503a.png "NCERT Exemplar solutions for Mathematics Exemplar [English] Class 12 chapter 7 - Integrals")
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Solutions for Chapter 7: Integrals
Below listed, you can find solutions for Chapter 7 of CBSE, Karnataka Board PUC NCERT Exemplar for Mathematics Exemplar [English] Class 12.
NCERT Exemplar solutions for Mathematics Exemplar [English] Class 12 7 Integrals Solved Examples [Pages 146 - 163]
Short Answer
Integrate `((2"a")/sqrt(x) - "b"/x^2 + 3"c"root(3)(x^2))` w.r.t. x
Evaluate `int (3"a"x)/("b"^2 + "c"^2x^2) "d"x`
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"`
Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1
Evaluate `int "dx"/sqrt((x - alpha)(beta - x)), beta > alpha`
Evaluate `int tan^8 x sec^4 x"d"x`
Find `int x^2/(x^4 + 3x^2 + 2) "d"x`
Find `int "dx"/(2sin^2x + 5cos^2x)`
Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
Find `int x^2tan^-1x"d"x`
Find `int sqrt(10 - 4x + 4x^2) "d"x`
Long Answer
Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
Objective Type Questions from 20 to 30
`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"`
`int "dx"/(sin^2x cos^2x)` is equal to ______.
tanx + cotx + C
x + cotx)2 + C
tanx – cotx + C
(tanx – cotx)2 + C
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`
`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`
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`
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
`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
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`
`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
`int (sin^6x)/(cos^8x) "d"x` = ______.
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
NCERT Exemplar solutions for Mathematics Exemplar [English] Class 12 7 Integrals Exercise [Pages 163 - 169]
Short Answer
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
Verify the following:
`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`
Evaluate the following:
`int ((x^2 + 2))/(x + 1) "d"x`
Evaluate the following:
`int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`
Evaluate the following:
`int ((1 + cosx))/(x + sinx) "d"x`
Evaluate the following:
`int ("d"x)/(1 + cos x)`
Evaluate the following:
`int tan^2x sec^4 x"d"x`
Evaluate the following:
`int (sinx + cosx)/sqrt(1 + sin 2x) "d"x`
Evaluate the following:
`int sqrt(1 + sinx)"d"x`
Evaluate the following:
`int x/(sqrt(x) + 1) "d"x` (Hint: Put `sqrt(x)` = z)
Evaluate the following:
`int sqrt(("a" + x)/("a" - x)) "d"x`
Evaluate the following:
`int x^(1/2)/(1 + x^(3/4)) "d"x` (Hint: Put `sqrt(x)` = z4)
Evaluate the following:
`int sqrt(1 + x^2)/x^4 "d"x`
Evaluate the following:
`int ("d"x)/sqrt(16 - 9x^2)`
Evaluate the following:
`int "dt"/sqrt(3"t" - 2"t"^2)`
Evaluate the following:
`int (3x - 1)/sqrt(x^2 + 9) "d"x`
Evaluate the following:
`int sqrt(5 - 2x + x^2) "d"x`
Evaluate the following:
`int x/(x^4 - 1) "d"x`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int sqrt(2"a"x - x^2) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
Evaluate the following:
`int (sin^6x + cos^6x)/(sin^2x cos^2x) "d"x`
Evaluate the following:
`int sqrt(x)/(sqrt("a"^3 - x^3)) "d"x`
Evaluate the following:
`int (cosx - cos2x)/(1 - cosx) "d"x`
Evaluate the following:
`int ("d"x)/(xsqrt(x^4 - 1))` (Hint: Put x2 = sec θ)
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
Evaluate the following:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Evaluate the following:
`int_1^2 ("d"x)/sqrt((x - 1)(2 - x))`
Evaluate the following:
`int_0^1 (x"d"x)/sqrt(1 + x^2)`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
Long Answer
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`
Evaluate the following:
`int sin^-1 sqrt(x/("a" + x)) "d"x` (Hint: Put x = a tan2θ)
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
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)
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
Objective Type Questions from 48 to 63
`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
`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"`
`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"`
`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"`
`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"`
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`
`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"`
`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"`
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
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
1
2
3
4
`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.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
`int (x + 3)/(x + 4)^2 "e"^x "d"x` = ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
`int sinx/(3 + 4cos^2x) "d"x` = ______.
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
Solutions for 7: Integrals
NCERT Exemplar solutions for Mathematics Exemplar [English] Class 12 chapter 7 - Integrals
Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics Exemplar [English] Class 12 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT Exemplar solutions for Mathematics Mathematics Exemplar [English] Class 12 CBSE, Karnataka Board PUC 7 (Integrals) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT Exemplar textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.
Concepts covered in Mathematics Exemplar [English] Class 12 chapter 7 Integrals are Definite Integrals, Evaluation of Definite Integrals by Substitution, Properties of Definite Integrals, Introduction of Integrals, Integration as an Inverse Process of Differentiation, Some Properties of Indefinite Integral, Methods of Integration> Integration by Substitution, Integration Using Trigonometric Identities, Integrals of Some Particular Functions, Overview of Integrals, Geometrical Interpretation of Indefinite Integrals, Methods of Integration> Integration Using Partial Fraction, Methods of Integration> Integration by Parts, Fundamental Theorem of Integral Calculus, Indefinite Integral Problems, Comparison Between Differentiation and Integration, Indefinite Integral by Inspection, Definite Integral as the Limit of a Sum, Evaluation of Simple Integrals of the Following Types and Problems.
Using NCERT Exemplar Mathematics Exemplar [English] Class 12 solutions Integrals exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Exemplar Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Mathematics Exemplar [English] Class 12 students prefer NCERT Exemplar Textbook Solutions to score more in exams.
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