NCERT Solutions for Class 12 Maths Chapter

 7 Integrals Recently updated for 2023-2024 .


 NCERT Solutions for Class 12 Maths Chapter 7

 Integrals



Chapter 7 Integrals


Class 12 SOLUTION And HANDWRITTEN NOTES FOR FREE


NCERT Solutions for Class 12 CBSE Maths – 

Chapter 7 Integrals

class 12 SOLUTION And HANDWRITTEN NOTES FOR FREE

NCERT Solutions for Class 12 CBSE Maths – 

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Chapter 7 Integrals


NCERT Solutions for Class 12 CBSE Maths – Chapter 7 Complete Questions solution of Maths chapter 7
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EX -7.1

Question 1.
sin 2x
Sol:




Question 2.
cos 3x
Sol:



Question 3.
e2x
Sol:



Question 4.
(ax + c)²
Sol:



Question 5.
sin2x−4e3x
Sol:


MathsQ 6.
∫(4e3x+1)dx
Sol:




Question 7.
∫x2(1−1x2)dx
Sol:



Question 8.
∫(ax2+bx+c)dx
Sol:


Question 9.
∫(2x2+ex)dx
Sol:



Question 10.
∫[x−−√−1x√]2dx
Sol:


Question 11.
∫x3+5x2−4x2dx
Sol: 



Ex 7.1 Class 12 MathsQ 12.
∫x3+3x+4x√dx
Sol:



Question 13.
∫x3−x2+x−1x−1dx
Sol:


Question 14.
∫(1−x)x−−√dx
Sol:



Question 15.
∫x−−√(3x2+2x+3)dx
Sol:



Question 16.
∫(2x−3cosx+ex)dx
Sol:



Question 17.
∫(2x2−3sinx+5x−−√)dx
Sol:



Question 18.
∫secx(secx+tanx)dx
Sol:



Question 19.
∫sec2xcosec2xdx
Sol:


MathsQ 20.
∫2−3sinxcos2xdx
Sol:



Question 21.
The antiderivative (x−−√+1x√) equals
(a) 13x13+2x12+c
(b) 23x23+12x2+c
(c) 23x32+2x12+c
(d) 32x32+12x12+c
Sol:




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
Sol:


Ex 7.2

MathsQ 1.
2x1+x2
Sol:




Question 2.
(logx)2x
Sol:



Question 3.
1x+xlogx
Sol:



Question 4.
sinx sin(cosx)
Sol:



Question 5.
sin(ax+b) cos(ax+b)
Sol:


Question 6.
ax+b−−−−−√
Sol:


Question 7.
xx+2−−−−√
Sol:



Question 8.
x1+2x2−−−−−−√
Sol:


Question 9.
(4x+2)x2+x+1−−−−−−−−√
Sol:



Question 10.
1x−x√
Sol:



Question 11.
xx+4√,x>0
Sol:



Question 12.
(x3−1)13.x5
Sol:


Question 13.
x2(2+3x3)3
Sol:



Question 14.
1x(logx)m,x>0
Sol:



Question 15.
x9−4x2
Sol:



Question 16.
e2x+3
Sol:



Question 17.
xex2
Sol:



Question 18.
etan−1x1+x2
Sol:




Question 19.
e2x−1e2x+1
Sol:




Question 20.
e2x−e2xe2x+e−2x
Sol:



Q 21.
tan²(2x-3)
Sol:
∫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

Question 22.
sec²(7-4x)
Sol:




Question 23.
sin−1x1−x2√
Sol:



Question 24.
2cosx−3sinx6cosx+4sinx
Sol:



Question 25.
1cos2x(1−tanx)2
Sol:




Question 26.
cosx√x√
Sol:




Q 27.
sin2x−−−−−√cos2x
Sol:





Question 28.
cosx1+sinx√
Sol:




Question 29.
cotx log sinx
Sol:



Question 30.
sinx1+cosx
Sol:



Question 31.
sinx(1+cosx)2
Sol:



Question 32.
11+cotx
Sol:


Question 33.
11−tanx
Sol:


Question 34.
tanx√sinxcosx
Sol:



Question 35.
(1+logx)2x
Sol:



Question 36.
(x+1)(x+logx)2x
Sol:



Question 37.
x3sin(tan−1x4)1+x8dx
Sol:



Question 38.
∫10x9+10xloge10x10+10xdx
(a) 10x – x10 + C
(b) 10x + x10 + C
(c) (10x – x10) + C
(d) log (10x + x10) + C
Sol:




Question 39.
∫dxsin2xcos2x=
(a) tanx + cotx + c
(b) tanx – cotx + c
(c) tanx cotx + c
(d) tanx – cot2x + c
Sol:


Ex 7.3


Question 1.
sin²(2x+5)
Sol:



Question 2.
 sin3x cos4x
 Sol:




Question 3.
∫cos2x cos4x cos6x dx
Sol:



Question 4.
∫sin3(2x+1)dx
Sol:



Question 5.
sin3x cos3x
Sol:



Question 6.
sinx sin2x sin3x
Sol:


Question 7.
sin 4x sin 8x
Sol:



Question 8.
1−cosx1+cosx
Sol:



Question 9.
cosx1+cosx
Sol:



Question 10.
∫sinx4 dx
Sol:


Q 11.
cos4 2x
Sol:




Question 12.
sin2x1+cosx
Sol:


Question 13.
cos2x−cos2αcosx−cosα
Sol:


Q 14.
cosx−sinx1+sin2x
Sol:


Question 15.
∫tan32xsec2xdx=I
Sol:



Question 16.
tan4x
Sol:



Question 17.
sin3x+cos3xsin2xcos2x
Sol:


Q 18.
cos2x+2sin2xcos2x
Sol:



 

Q 19.
1sinxcos3x
Sol:




Question 20.
cos2x(cosx+sinx)2
Sol:





Q 21.
sin-1 (cos x)
Sol:







Q 22.
∫1cos(x−a)cos(x−b)dx
Sol:




Question 23.
∫sin2x−cos2xsin2xcos2xdxisequalto
(a) tanx+cotx+c
(b) tanx+cosecx+c
(c) -tanx+cotx+c
(d) tanx+secx+c
Sol:




Question 24.
∫ex(1+x)cos2(ex.x)dxisequalto
(a) -cot(e.xx)+c
(b) tan(xex)+c
(c) tan(ex)+c
(d) cot ex+c
Sol:


 

Ex 7.4

Question 1.
3x2x6+1
Sol:




Question 2.
11+4x2√
Sol:



Question 3.
1(2−x)2+1√
Sol:


Q 4.
19−25x2√
Sol:



Question 5.
3x1+2x4
Sol:



Question 6.
x21−x6
Sol:




Question 7.
x−1x2−1√
Sol:





Question 8.
x2x6+a6√
Sol:



Question 9.
sec2xtan2x+4√
Sol:



Question 10.
1x2+2x+2√
Sol:



Question 11.

19x2+6x+5
Sol:




Ex 7.4 Class 12 MathsQ 12.
17−6x−x2√
Sol:



Question 13.
1(x−1)(x−2)√
Sol: 



Question 14.
18+3x−x2√
Sol:



Question 15.
1(x−a)(x−b)√
Sol:


Question 16.
4x+12x2+x−3√
Sol:



Question 17.
x+2x2−1√
Sol:




Question 18.
5x−21+2x+3x2
Sol:


put 5x-2=Addx(1+2x+3x²)+B
⇒ 6A=5, A=56−2=2A+B, B=−113

Question 19.
6x+7(x−5)(x−4)√
Sol:







Question 20.
x+24x−x2√
Sol:






Q 21.
x+2x2+2x+3√
Sol:




Question 22.
x+3x2−2x−5
Sol:




Q 23.
5x+3x2+4x+10√
Sol:





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
Sol:



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
Sol: 


Ex 7.5

Question 1.
x(x+1)(x+2)
Sol:




Question 2.
1x2−9
Sol:



Question 3.
3x−1(x−1)(x−2)(x−3)
Sol:



Question 4.
x(x−1)(x−2)(x−3)
Sol:


Question 5.
2xx2+3x+2
Sol:


Question 6.
1−x2x(1−2x)
Sol:



Question 7.
x(x2+1)(x−1)
Sol:




Question 8.
x(x−1)2(x+2)
Sol:


Question 9.
3x+5x3−x2−x+1
Sol:


Question 10.
2x−3(x2−1)(2x+3)
Sol:



Question 11.
5x(x−1)(x2−4)
Sol:


Question 12.
x3+x+1x2−1
Sol:



Question 13.
2(1−x)(1+x2)
Sol:



Q14.
3x−1(x+2)2
Sol:



Q15.
1x4−1
Sol:



Q16.
1x(xn+1)
[Hint : multiply numerator and denominator by xn-1 and put xn = t ]
Sol:





Q17.
cosx(1−sinx)(2−sinx)
Sol:



Q18.
(x2+1)(x2+2)(x2+3)(x2+4)
Sol:





Q19.
2x(x2+1)(x2+3)
Sol:


Q20.
1x(x4−1)
Sol:


Q21.
1ex−1
Sol:



Q22.
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
Sol:



Q23.
∫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
Sol:


Ex 7.6

Q 1.
x sinx
Sol:




Q 2.
x sin3x
Sol:


Q 3.
x2ex
Sol:


Q 4.
x logx
Sol:


Q 5.
x log2x
Sol:



Q 6.
x2logx
Sol:


Q 7.
xsin−1x
Sol:


Q 8.
xtan−1x
Sol:



Q 9.
xcos−1x
Sol:




Q 10.
(sin−1x)2
Sol:


Q 11.
xcos−1x1−x2√
Sol:



Q 12.
x sec²x
Sol:



Q 13.
tan−1x
Sol:


Q 14.
x(logx)²
Sol:


Q 15.
(x²+1)logx
Sol:




Q 16.
ex(sinx+cosx)
Sol:


Q 17.
xex(1+x)2
Sol:


Q 18.
ex(1+sinx)1+cosx
Sol:




Q 19.
ex(1x−1x2)
Sol:


Q 20.
(x−2)ex(x−1)3
Sol:



Q 21.
e2xsinx
Sol:



Q 22.
sin−1(2x1+x2)
Sol:


Choose the correct answer in exercise 23 and 24

Q 23.
∫x2ex3dxequals
(a) 13ex3+c
(b) 13+ex2+c
(c) 12ex3+c
(d) 12ex2+c
Sol:




Q 24.
∫exsecx(1+tanx)dxequals
(a) excosx+c
(b) exsecx+c
(c) exsinx+c
(d) extanx+c
Sol:


Ex 7.7

Q 1.
4−x2−−−−−√
Sol:



Q 2.
1−4x2−−−−−−√
Sol:



Q 3.
x2+4x+6−−−−−−−−−√
Sol:



Q 4.
x2+4x+1−−−−−−−−−√
Sol:


Q 5.
1−4x−x2−−−−−−−−−√
Sol:


Q 6.
x2+4x−5−−−−−−−−−√
Sol:



Q 7.
1+3x−x2−−−−−−−−−√
Sol:


Q 8.
x2+3x−−−−−−√
Sol:



Q 9.
1+x29−−−−−√
Sol:



Choose the correct answer in the Exercises 10 to 11:

Q 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
Sol:




Q 11.
∫x2−8x+7−−−−−−−−−√dx
Sol:


Ex 7.8

Q 1.
∫baxdx
Sol:




Q 2.
∫50(x+1)dx
Sol:




Q 3.
∫32x2dx
Sol:





Q 4.
∫41(x2−x)dx
Sol:





Q 5.
∫1−1exdx
Sol:



Q 6.
∫40(x+e2x)dx
Sol:



Ex 7.9

Q 1.
∫1−1(x+1)dx
Sol:



Q 2.
∫321xdx
Sol:



Q 3.
∫21(4x3−5x2+6x+9)dx
Sol:



Q 4.
∫π40sin2xdx
Sol:



Q 5.
∫π20cos2xdx
Sol:



Q 6.
∫54exdx
Sol:



Q 7.
∫π40tanxdx
Sol:



Q 8.
∫π4π6cosecxdx
Sol:



Q 9.
∫10dx1−x2√
Sol:



Q 10.
∫10dx1+x2
Sol:



Q 11.
∫32dxx2−1
Sol:



Q 12.
∫π20cos2xdx
Sol:



Q 13.
∫32xx2+1dx
Sol:



Q 14.
∫102x+35x2+1dx
Sol:



Q 15.
∫10xex2dx
Sol:



Q 16.
∫215x2x2+4x+3dx
Sol:




Q 17.
∫π40(2sec2x+x3+2)dx
Sol:



Q 18.
∫π0(sin2x2−cos2x2)dx
Sol:



Q 19.
∫206x+3x2+4dx
Sol:



Q 20.
∫10(xex+sinπx4)dx
Sol:



Q 21.
∫3√1dx1+x2equals
(a) π3
(b) 2π3
(c) π6
(d) π12
Sol:



Q 22.
∫230dx4+9x2equals
(a) π6
(b) π12
(c) π24
(d) π4
Sol:



 

 

Ex 7.10

Q 1:




Sol:



Q 2:



Sol:



Q 3:



Sol:



Q 4:



Sol:



Q 5:



Sol:





Q 6:



Sol:





Q 7:



Sol:



Q 8:



Sol:



Q 9:



Sol:



Q 10:



Sol:



Q 11:



Sol:









Q 12:



Sol:







Q 13:



Sol:



Q 14:



Sol:



Q 15:



Sol:



Q 16:



Sol:



Q 17:



Sol:





Q 18:


Sol:





Q 19:




Sol:



Q 20 



Sol: