The topics and sub-topics included in the Integrals chapter are the following:

NCERT Solutions for Class 12 Maths Chapter 7  Integrals: Engineering aspirants and students appearing for CBSE Class 12 board exams must take the NCERT Mathematics textbooks seriously and finish them from top to bottom. They must also go through the NCERT solutions for Class 12 Maths and Class 11 Maths to have a better understanding of the various concepts. In this article, we will provide you with NCERT Solutions for Class 12 Maths Chapter 7 – Integrals which have been designed by the best teachers in India.

Free download NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11, Ex 7.10, Ex 7.9, Ex 7.8, Ex 7.7, Ex 7.6, Ex 7.5, Ex 7.4, Ex 7.3, Ex 7.2, Ex 7.1 Integrals PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

Class 12 Maths Chapter 7 NCERT Solutions Integrals

NCERT Solutions for Class 12 Maths Chapter 7 – Integrals contains step-by-step and detailed solutions for every question.

1. Introduction
2. Integration as an Inverse Process of Differentiation
3. Geometrical interpretation of indefinite integral
4. Some properties of indefinite integral
5. Comparison between differentiation and integration
6. Methods of Integration
7. Integration by substitution
8. Integration using trigonometric identities
9. Integrals of Some Particular Functions
10. Integration by Partial Fractions
11. Integration by Parts
12. Integral of the type
13. Integrals of some more types
14. Definite Integral
15. Definite integral as the limit of a sum
16. Fundamental Theorem of Calculus
17. Area function
18. First fundamental theorem of integral calculus
19. Second fundamental theorem of integral calculus
20. Evaluation of Definite Integrals by Substitution
21. Some Properties of Definite Integrals

Integrals Ex 7.1 Class 12

Ex 7.1 Class 12 Maths Question 1.
sin 2x
Solution:

Ex 7.1 Class 12 Maths Question 2.
cos 3x
Solution:

Ex 7.1 Class 12 Maths Question 3.
e2x
Solution:

Ex 7.1 Class 12 Maths Question 4.
(ax + c)²
Solution:

Ex 7.1 Class 12 Maths Question 5.
sin2x−4e3x
Ex 7.1 Class 12 Maths Solution:

Find the following integrals in Exercises 6 to 20 :

Ex 7.1 Class 12 Maths Question 6.
∫(4e3x+1)dx
Solution:

Ex 7.1 Class 12 Maths Question 7.
∫x2(1−1x2)dx
Solution:

Ex 7.1 Class 12 Maths Question 8.
∫(ax2+bx+c)dx
Solution:

Ex 7.1 Class 12 Maths Question 9.
∫(2x2+ex)dx
Solution:

Ex 7.1 Class 12 Maths Question 10.
∫[x−−√−1x√]2dx
Solution:

Ex 7.1 Class 12 Maths Question 11.
∫x3+5x2−4x2dx
Solution:

Ex 7.1 Class 12 Maths Question 12.
∫x3+3x+4x√dx
Solution:

Ex 7.1 Class 12 Maths Question 13.
∫x3−x2+x−1x−1dx
Solution:

Ex 7.1 Class 12 Maths Question 14.
∫(1−x)x−−√dx
Solution:

Ex 7.1 Class 12 Maths Question 15.
∫x−−√(3x2+2x+3)dx
Solution:

Ex 7.1 Class 12 Maths Question 16.
∫(2x−3cosx+ex)dx
Solution:

Ex 7.1 Class 12 Maths Question 17.
∫(2x2−3sinx+5x−−√)dx
Solution:

Ex 7.1 Class 12 Maths Question 18.
∫secx(secx+tanx)dx
Solution:

Ex 7.1 Class 12 Maths Question 19.
∫sec2xcosec2xdx
Solution:

Ex 7.1 Class 12 Maths Question 20.
∫2−3sinxcos2xdx
Solution:

Choose the correct answer in Exercises 21 and 22.
Ex 7.1 Class 12 Maths Question 21.
The antiderivative (x−−√+1x√) equals
(a) 13x13+2x12+c
(b) 23x23+12x2+c
(c) 23x32+2x12+c
(d) 32x32+12x12+c
Solution:

Ex 7.1 Class 12 Maths Question 22.
If ddxf(x)=4x3−3x4 such that f(2)=0 then f(x) is
(a) x4+1x3−1298
(b) x3+1x4+1298
(c) x4+1x3+1298
(d) x3+1x4−1298
Solution:

Integrals Ex 7.2 Class 12

Integrate the functions in Exercises 1 to 37:

Ex 7.2 Class 12 Maths Question 1.
2x1+x2
Solution:

Ex 7.2 Class 12 Maths Question 2.
(logx)2x
Solution:

Ex 7.2 Class 12 Maths Question 3.
1x+xlogx
Solution:

Ex 7.2 Class 12 Maths Question 4.
sinx sin(cosx)
Solution:

Ex 7.2 Class 12 Maths Question 5.
sin(ax+b) cos(ax+b)
Solution:

Ex 7.2 Class 12 Maths Question 6.
ax+b−−−−−√
Solution:

Ex 7.2 Class 12 Maths Question 7.
xx+2−−−−√
Solution:

Ex 7.2 Class 12 Maths Question 8.
x1+2x2−−−−−−√
Solution:

Ex 7.2 Class 12 Maths Question 9.
(4x+2)x2+x+1−−−−−−−−√
Solution:

Ex 7.2 Class 12 Maths Question 10.
1x−x√
Solution:

Ex 7.2 Class 12 Maths Question 11.
xx+4√,x>0
Solution:

Ex 7.2 Class 12 Maths Question 12.
(x3−1)13.x5
Solution:

Ex 7.2 Class 12 Maths Question 13.
x2(2+3x3)3
Solution:

Ex 7.2 Class 12 Maths Question 14.
1x(logx)m,x>0
Solution:

Ex 7.2 Class 12 Maths Question 15.
x9−4x2
Solution:

Ex 7.2 Class 12 Maths Question 16.
e2x+3
Solution:

Ex 7.2 Class 12 Maths Question 17.
xex2
Solution:

Ex 7.2 Class 12 Maths Question 18.
etan−1x1+x2
Solution:

Ex 7.2 Class 12 Maths Question 19.
e2x−1e2x+1
Solution:

Ex 7.2 Class 12 Maths Question 20.
e2x−e2xe2x+e−2x
Solution:

Ex 7.2 Class 12 Maths Question 21.
tan²(2x-3)
Solution:
∫tan²(2x-3)dx = ∫[sec²(2x-3)-1]dx = I
put 2x-3 = t
so that 2dx = dt
I = 12 ∫sec²t dt-x+c
= 12t−x+c
= 12tan(2x−3)−x+c
Ex 7.2 Class 12 Maths Question 22.
sec²(7-4x)
Solution:

Ex 7.2 Class 12 Maths Question 23.
sin−1x1−x2√
Solution:

Ex 7.2 Class 12 Maths Question 24.
2cosx−3sinx6cosx+4sinx
Solution:

Ex 7.2 Class 12 Maths Question 25.
1cos2x(1−tanx)2
Solution:

Ex 7.2 Class 12 Maths Question 26.
cosx√x√
Solution:

Ex 7.2 Class 12 Maths Question 27.
sin2x−−−−−√cos2x
Solution:

Ex 7.2 Class 12 Maths Question 28.
cosx1+sinx√
Solution:

Ex 7.2 Class 12 Maths Question 29.
cotx log sinx
Solution:

Ex 7.2 Class 12 Maths Question 30.
sinx1+cosx
Solution:

Ex 7.2 Class 12 Maths Question 31.
sinx(1+cosx)2
Solution:

Ex 7.2 Class 12 Maths Question 32.
11+cotx
Solution:

Ex 7.2 Class 12 Maths Question 33.
11−tanx
Solution:

Ex 7.2 Class 12 Maths Question 34.
tanx√sinxcosx
Solution:

Ex 7.2 Class 12 Maths Question 35.
(1+logx)2x
Solution:

Ex 7.2 Class 12 Maths Question 36.
(x+1)(x+logx)2x
Solution:

Ex 7.2 Class 12 Maths Question 37.
x3sin(tan−1x4)1+x8dx
Solution:

Choose the correct answer in exercises 38 and 39

Ex 7.2 Class 12 Maths Question 38.
∫10x9+10xloge10x10+10xdx
(a) 10x – x10 + C
(b) 10x + x10 + C
(c) (10x – x10) + C
(d) log (10x + x10) + C
Solution:

Ex 7.2 Class 12 Maths Question 39.
∫dxsin2xcos2x=
(a) tanx + cotx + c
(b) tanx – cotx + c
(c) tanx cotx + c
(d) tanx – cot2x + c
Solution:

Integrals Ex 7.3 Class 12

Find the integrals of the functions in Exercises 1 to 22.
Ex 7.3 Class 12 Maths Question 1.
sin²(2x+5)
Solution:

Ex 7.3 Class 12 Maths Question 2.
sin3x cos4x
Solution:

Ex 7.3 Class 12 Maths Question 3.
∫cos2x cos4x cos6x dx
Solution:

Ex 7.3 Class 12 Maths Question 4.
∫sin³(2x+1)dx
Solution:

Ex 7.3 Class 12 Maths Question 5.
sin³x cos³x
Solution:

Ex 7.3 Class 12 Maths Question 6.
sinx sin2x sin3x
Solution:

Ex 7.3 Class 12 Maths Question 7.
sin 4x sin 8x
Solution:

Ex 7.3 Class 12 Maths Question 8.
1−cosx1+cosx
Solution:

Ex 7.3 Class 12 Maths Question 9.
cosx1+cosx
Solution:

Ex 7.3 Class 12 Maths Question 10.
∫sinx⁴ dx
Solution:

Ex 7.3 Class 12 Maths Question 11.
cos⁴ 2x
Solution:

Ex 7.3 Class 12 Maths Question 12.
sin2x1+cosx
Solution:

Ex 7.3 Class 12 Maths Question 13.
cos2x−cos2αcosx−cosα
Solution:

Ex 7.3 Class 12 Maths Question 14.
cosx−sinx1+sin2x
Solution:

Ex 7.3 Class 12 Maths Question 15.
∫tan32xsec2xdx=I
Solution:

Ex 7.3 Class 12 Maths Question 16.
tan⁴x
Solution:

Ex 7.3 Class 12 Maths Question 17.
sin3x+cos3xsin2xcos2x
Solution:

Ex 7.3 Class 12 Maths Question 18.
cos2x+2sin2xcos2x
Solution:

Ex 7.3 Class 12 Maths Question 19.
1sinxcos3x
Solution:

Ex 7.3 Class 12 Maths Question 20.
cos2x(cosx+sinx)2
Solution:

Ex 7.3 Class 12 Maths Question 21.
sin^-1 (cos x)
Solution:


Ex 7.3 Class 12 Maths Question 22.
∫1cos(x−a)cos(x−b)dx
Solution:

Ex 7.3 Class 12 Maths Question 23.
∫sin2x−cos2xsin2xcos2xdxisequalto
(a) tanx+cotx+c
(b) tanx+cosecx+c
(c) -tanx+cotx+c
(d) tanx+secx+c
Solution:

Ex 7.3 Class 12 Maths Question 24.
∫ex(1+x)cos2(ex.x)dxisequalto
(a) -cot(e.x^x)+c
(b) tan(xe^x)+c
(c) tan(e^x)+c
(d) cot e^x+c
Solution:

Integrals Class 12 Ex 7.4

Ex 7.4 Class 12 Maths Question 1.
3x2x6+1
Solution:

Ex 7.4 Class 12 Maths Question 2.
11+4x2√
Solution:

Ex 7.4 Class 12 Maths Question 3.
1(2−x)2+1√
Solution:

Ex 7.4 Class 12 Maths Question 4.
19−25x2√
Solution:

Ex 7.4 Class 12 Maths Question 5.
3x1+2x4
Solution:

Ex 7.4 Class 12 Maths Question 6.
x21−x6
Solution:

Ex 7.4 Class 12 Maths Question 7.
x−1x2−1√
Solution:

Ex 7.4 Class 12 Maths Question 8.
x2x6+a6√
Solution:

Ex 7.4 Class 12 Maths Question 9.
sec2xtan2x+4√
Solution:

Ex 7.4 Class 12 Maths Question 10.
1x2+2x+2√
Solution:

Ex 7.4 Class 12 Maths Question 11.

19x2+6x+5
Solution:

Ex 7.4 Class 12 Maths Question 12.
17−6x−x2√
Solution:

Ex 7.4 Class 12 Maths Question 13.
1(x−1)(x−2)√
Solution:

Ex 7.4 Class 12 Maths Question 14.
18+3x−x2√
Solution:

Ex 7.4 Class 12 Maths Question 15.
1(x−a)(x−b)√
Solution:

Ex 7.4 Class 12 Maths Question 16.
4x+12x2+x−3√
Solution:

Ex 7.4 Class 12 Maths Question 17.
x+2x2−1√
Solution:


Ex 7.4 Class 12 Maths Question 18.
5x−21+2x+3x2
Solution:
put 5x-2=Addx(1+2x+3x²)+B
⇒ 6A=5, A=56−2=2A+B, B=−113

Ex 7.4 Class 12 Maths Question 19.
6x+7(x−5)(x−4)√
Solution:



Ex 7.4 Class 12 Maths Question 20.
x+24x−x2√
Solution:



Ex 7.4 Class 12 Maths Question 21.
x+2x2+2x+3√
Solution:


Ex 7.4 Class 12 Maths Question 22.
x+3x2−2x−5
Solution:


Ex 7.4 Class 12 Maths Question 23.
5x+3x2+4x+10√
Solution:


Ex 7.4 Class 12 Maths Question 24.
∫dxx2+2x+2equals
(a) xtan^-1(x+1)+c
(b) (x+1)tan^-1x+c
(c) tan^-1(x+1)+c
(d) tan^-1x+c
Solution:

Ex 7.4 Class 12 Maths Question 25.
∫dx9x−4x2√equals
(a) 19sin−1(9x−88)+c
(b) 12sin−1(8x−99)+c
(c) 13sin−1(9x−88)+c
(d) sin−1(9x−89)+c
Solution:

Integrals Class 12 Ex 7.5

Ex 7.5 Class 12 Maths Question 1.
x(x+1)(x+2)
Solution:

Ex 7.5 Class 12 Maths Question 2.
1x2−9
Solution:

Ex 7.5 Class 12 Maths Question 3.
3x−1(x−1)(x−2)(x−3)
Solution:

Ex 7.5 Class 12 Maths Question 4.
x(x−1)(x−2)(x−3)
Solution:

Ex 7.5 Class 12 Maths Question 5.
2xx2+3x+2
Solution:

Ex 7.5 Class 12 Maths Question 6.
1−x2x(1−2x)
Solution:

Ex 7.5 Class 12 Maths Question 7.
x(x2+1)(x−1)
Solution:


Ex 7.5 Class 12 Maths Question 8.
x(x−1)2(x+2)
Solution:

Ex 7.5 Class 12 Maths Question 9.
3x+5x3−x2−x+1
Solution:

Ex 7.5 Class 12 Maths Question 10.
2x−3(x2−1)(2x+3)
Solution:

Ex 7.5 Class 12 Maths Question 11.
5x(x−1)(x2−4)
Solution:

Ex 7.5 Class 12 Maths Question 12.
x3+x+1x2−1
Solution:

Ex 7.5 Class 12 Maths Question 13.
2(1−x)(1+x2)
Solution:

Ex 7.5 Class 12 Maths Question 14.
3x−1(x+2)2
Solution:

Ex 7.5 Class 12 Maths Question 15.
1x4−1
Solution:

Ex 7.5 Class 12 Maths Question 16.
1x(xn+1)
[Hint : multiply numerator and denominator by x^n-1 and put xⁿ = t ]
Solution:


Ex 7.5 Class 12 Maths Question 17.
cosx(1−sinx)(2−sinx)
Solution:

Ex 7.5 Class 12 Maths Question 18.
(x2+1)(x2+2)(x2+3)(x2+4)
Solution:


Ex 7.5 Class 12 Maths Question 19.
2x(x2+1)(x2+3)
Solution:

Ex 7.5 Class 12 Maths Question 20.
1x(x4−1)
Solution:

Ex 7.5 Class 12 Maths Question 21.
1ex−1
Solution:

Ex 7.5 Class 12 Maths Question 22.
choose the correct answer in each of the following :
∫xdx(x−1)(x−2)equals
(a) log∣∣∣(x−1)2x−2∣∣∣+c
(b) log∣∣∣(x−2)2x−1∣∣∣+c
(c) log∣∣(x−12x−2)∣∣+c
(d) log|(x-1)(x-2)|+c
Solution:

Ex 7.5 Class 12 Maths Question 23.
∫dxx(x2+1)equals
(a) log|x|−12log(x2+1)+c
(b) log|x|+12log(x2+1)+c
(c) −log|x|+12log(x2+1)+c
(d) 12log|x|+log(x2+1)+c
Solution:

Integrals Class 12 Ex 7.6

Ex 7.6 Class 12 Maths Question 1.
x sinx
Solution:

Ex 7.6 Class 12 Maths Question 2.
x sin3x
Solution:

Ex 7.6 Class 12 Maths Question 3.
x2ex
Solution:

Ex 7.6 Class 12 Maths Question 4.
x logx
Solution:

Ex 7.6 Class 12 Maths Question 5.
x log2x
Solution:

Ex 7.6 Class 12 Maths Question 6.
x2logx
Solution:

Ex 7.6 Class 12 Maths Question 7.
xsin−1x
Solution:

Ex 7.6 Class 12 Maths Question 8.
xtan−1x
Solution:

Ex 7.6 Class 12 Maths Question 9.
xcos−1x
Solution:


Ex 7.6 Class 12 Maths Question 10.
(sin−1x)2
Solution:

Ex 7.6 Class 12 Maths Question 11.
xcos−1x1−x2√
Solution:

Ex 7.6 Class 12 Maths Question 12.
x sec²x
Solution:

Ex 7.6 Class 12 Maths Question 13.
tan−1x
Solution:

Ex 7.6 Class 12 Maths Question 14.
x(logx)²
Solution:

Ex 7.6 Class 12 Maths Question 15.
(x²+1)logx
Solution:


Ex 7.6 Class 12 Maths Question 16.
ex(sinx+cosx)
Solution:

Ex 7.6 Class 12 Maths Question 17.
xex(1+x)2
Solution:

Ex 7.6 Class 12 Maths Question 18.
ex(1+sinx)1+cosx
Solution:


Ex 7.6 Class 12 Maths Question 19.
ex(1x−1x2)
Solution:

Ex 7.6 Class 12 Maths Question 20.
(x−2)ex(x−1)3
Solution:

Ex 7.6 Class 12 Maths Question 21.
e2xsinx
Solution:

Ex 7.6 Class 12 Maths Question 22.
sin−1(2x1+x2)
Solution:

Choose the correct answer in exercise 23 and 24

Ex 7.6 Class 12 Maths Question 23.
∫x2ex3dxequals
(a) 13ex3+c
(b) 13+ex2+c
(c) 12ex3+c
(d) 12ex2+c
Solution:

Ex 7.6 Class 12 Maths Question 24.
∫exsecx(1+tanx)dxequals
(a) excosx+c
(b) exsecx+c
(c) exsinx+c
(d) extanx+c
Solution:

Class 12 Integrals Ex 7.7

Integral the function in exercises 1 to 9

Ex 7.7 Class 12 Maths Question 1.
4−x2−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 2.
1−4x2−−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 3.
x2+4x+6−−−−−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 4.
x2+4x+1−−−−−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 5.
1−4x−x2−−−−−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 6.
x2+4x−5−−−−−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 7.
1+3x−x2−−−−−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 8.
x2+3x−−−−−−√
Solution:

Ex 7.7 Class 12 Maths Question 9.
1+x29−−−−−√
Solution:

Choose the correct answer in the Exercises 10 to 11:

Ex 7.7 Class 12 Maths Question 10.
∫1+x2−−−−−√dxisequalto
(a) x21+x2−−−−−√+12log|x+1+x2−−−−−√|+c
(b) 23(1+x2)32+c
(c) 23x(1+x2)32+c
(d) x221+x2−−−−−√+12x2log∣∣x+1+x2−−−−−√∣∣+c
Solution:

Ex 7.7 Class 12 Maths Question 11.
∫x2−8x+7−−−−−−−−−√dxisequalto

Solution:

Class 12 Integrals Ex 7.8

Ex 7.8 Class 12 Maths Question 1.
∫baxdx
Solution:

Ex 7.8 Class 12 Maths Question 2.
∫50(x+1)dx
Solution:

Ex 7.8 Class 12 Maths Question 3.
∫32x2dx
Solution:

Ex 7.8 Class 12 Maths Question 4.
∫41(x2−x)dx
Solution:

Ex 7.8 Class 12 Maths Question 5.
∫1−1exdx
Solution:

Ex 7.8 Class 12 Maths Question 6.
∫40(x+e2x)dx
Solution:

Class 12 Integrals Ex 7.9

Ex 7.9 Class 12 Maths Question 1.
∫1−1(x+1)dx
Solution:

Ex 7.9 Class 12 Maths Question 2.
∫321xdx
Solution:

Ex 7.9 Class 12 Maths Question 3.
∫21(4x3−5x2+6x+9)dx
Solution:

Ex 7.9 Class 12 Maths Question 4.
∫π40sin2xdx
Solution:

Ex 7.9 Class 12 Maths Question 5.
∫π20cos2xdx
Solution:

Ex 7.9 Class 12 Maths Question 6.
∫54exdx
Solution:

Ex 7.9 Class 12 Maths Question 7.
∫π40tanxdx
Solution:

Ex 7.9 Class 12 Maths Question 8.
∫π4π6cosecxdx
Solution:

Ex 7.9 Class 12 Maths Question 9.
∫10dx1−x2√
Solution:

Ex 7.9 Class 12 Maths Question 10.
∫10dx1+x2
Solution:

Ex 7.9 Class 12 Maths Question 11.
∫32dxx2−1
Solution:

Ex 7.9 Class 12 Maths Question 12.
∫π20cos2xdx
Solution:

Ex 7.9 Class 12 Maths Question 13.
∫32xx2+1dx
Solution:

Ex 7.9 Class 12 Maths Question 14.
∫102x+35x2+1dx
Solution:

Ex 7.9 Class 12 Maths Question 15.
∫10xex2dx
Solution:

Ex 7.9 Class 12 Maths Question 16.
∫215x2x2+4x+3dx
Solution:

Ex 7.9 Class 12 Maths Question 17.
∫π40(2sec2x+x3+2)dx
Solution:

Ex 7.9 Class 12 Maths Question 18.
∫π0(sin2x2−cos2x2)dx
Solution:

Ex 7.9 Class 12 Maths Question 19.
∫206x+3x2+4dx
Solution:

Ex 7.9 Class 12 Maths Question 20.
∫10(xex+sinπx4)dx
Solution:

Ex 7.9 Class 12 Maths Question 21.
∫3√1dx1+x2equals
(a) π3
(b) 2π3
(c) π6
(d) π12
Solution:

Ex 7.9 Class 12 Maths Question 22.
∫230dx4+9x2equals
(a) π6
(b) π12
(c) π24
(d) π4
Solution:

Integration Class 12 Ex 7.10

Ex 7.10 Class 12 Maths Question 1:

Solution:

Ex 7.10 Class 12 Maths Question 2:

Solution:

Ex 7.10 Class 12 Maths Question 3:

Solution:

Ex 7.10 Class 12 Maths Question 4:

Solution:

Ex 7.10 Class 12 Maths Question 5:

Solution:


Ex 7.10 Class 12 Maths Question 6:

Solution:


Ex 7.10 Class 12 Maths Question 7:

Solution:

Ex 7.10 Class 12 Maths Question 8:

Solution:

Ex 7.10 Class 12 Maths Question 9:

Solution:

Ex 7.10 Class 12 Maths Question 10:

Solution:

Ex 7.10 Class 12 Maths Question 11:

Solution:




Ex 7.10 Class 12 Maths Question 12:

Solution:



Ex 7.10 Class 12 Maths Question 13:

Solution:

Ex 7.10 Class 12 Maths Question 14:

Solution:

Ex 7.10 Class 12 Maths Question 15:

Solution:

Ex 7.10 Class 12 Maths Question 16:

Solution:

Ex 7.10 Class 12 Maths Question 17:

Solution:


Ex 7.10 Class 12 Maths Question 18:

Solution:


Ex 7.10 Class 12 Maths Question 19:


Solution:

Ex 7.10 Class 12 Maths Question 20:


Solution:

NCERT Solutions for Class 2 Maths Integration Class 12 Ex 7.11

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

Miscellaneous Exercise Class 11 Maths Question 1:

Solution:

Miscellaneous Exercise Class 11 Maths Question 2:

Solution:

Miscellaneous Exercise Class 11 Maths Question 3:

Solution:

vMiscellaneous Exercise Class 11 Maths Question 4:

Solution:

Miscellaneous Exercise Class 11 Maths Question 5:

Solution:

Miscellaneous Exercise Class 11 Maths Question 6:

Solution:

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Integration Formulas

गणित कक्षा 12 समाकलन प्रश्नावली 7.1

निम्नलिखित फलनों के प्रतिअवकलज ( समाकलन) निरीक्षण विधि द्वारा ज्ञात कीजिए।

Ex 7.1 Class 11 Maths प्रश्न 1.
sin 2x
हल-
∫sin 2x dx
हम जानते हैं कि, ddx cos2x = -2 sin 2x
या sin 2x = −12ddxcos2x
sin 2x = ddx(−12cos2x)
अतः sin 2x का प्रति अवकलज −12cos 2x है।

Ex 7.1 Class 11 Maths प्रश्न 2.
cos 3x
हल-
∫cos 3x dx
हम जानते हैं कि, ddx sin 3x = 3 cos 3x

अत: cos3x का प्रति अवकलज 13 sin 3x हैं।

Ex 7.1 Class 11 Maths प्रश्न 3.
e^2x
हल-
∫e^2x dx
हम जानते हैं कि,

Ex 7.1 Class 11 Maths प्रश्न 4.
(ax + b)²
हल-
∫(ax + b)² dx
हम जानते हैं कि,

Ex 7.1 Class 11 Maths प्रश्न 5.
sin 2x – 4e^3x
हल-
हम जानते हैं

निम्नलिखित समाकलनों को ज्ञात कीजिए।

Ex 7.1 Class 11 Maths प्रश्न 6.
∫(4e^3x + 1) dx
हल-
∫(4e^3x + 1) dx = 4∫e^3xdx + ∫1 dx

Ex 7.1 Class 11 Maths प्रश्न 7.

हल-

Ex 7.1 Class 11 Maths प्रश्न 8.
∫(ax² + bx + c) dx
हल-
∫(ax² + bx + c) dx = a∫x²dx + b∫xdx + c∫1 dx

Ex 7.1 Class 11 Maths प्रश्न 9.
∫(2x² + e^x) dx
हल-
∫(2x² + e^x) dx = 2∫x² dx + ∫e^x dx

Ex 7.1 Class 11 Maths प्रश्न 10.

हल-

Ex 7.1 Class 11 Maths प्रश्न 11.

हल-

Ex 7.1 Class 11 Maths प्रश्न 12.

हल-

Ex 7.1 Class 11 Maths प्रश्न 13.

हल-

Ex 7.1 Class 11 Maths प्रश्न 14.
∫(1-x)√x dx
हल-
∫(1-x)√x dx
=∫(√x-x√x)dx

Ex 7.1 Class 11 Maths प्रश्न 15.
∫√x(3x² + 2x + 3)dx
हल-
∫√x(3x² + 2x + 3)dx

Ex 7.1 Class 11 Maths प्रश्न 16.
∫(2x – 3cosx + e^x)dx
हल-
∫(2x – 3cosx + e^x)dx

प्रश्न 17.
∫(2x² – 3sinx + 5√x)dx
हल-
∫(2x² – 3sinx + 5√x)dx
= 2∫x² dx – 3 ∫ sinx dx + 5∫√x dx

Ex 7.1 Class 11 Maths प्रश्न 18.
∫secx (sec x + tan x) dx
हल-
∫sec x (sec x + tan x) dx
= ∫(sec² x + sec x tan x) dx
= ∫(sec² x dx
= ∫sec x tan x dx
tan x + sec x + C

Ex 7.1 Class 11 Maths प्रश्न 19.

हल-

Ex 7.1 Class 11 Maths प्रश्न 20.

हल-

प्रश्न 21 एवं 22 में सही उत्तर का चयन कीजिए

Ex 7.1 Class 11 Maths प्रश्न 21

हल-

Ex 7.1 Class 11 Maths प्रश्न 22

हल-