Chapter 7: 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(1−1×2)dx
Solution:

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

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

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

Ex 7.1 Class 12 Maths Question 11.
∫x3+5×2−4x2dx
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−−√(3×2+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.
∫(2×2−3sinx+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) 13×13+2×12+c
(b) 23×23+12×2+c
(c) 23×32+2×12+c
(d) 32×32+12×12+c
Solution:

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

Move to Top → Class 12 Integrals Chapter 7

Integrals Ex 7.2 Class 12

Integrate the functions in Exercises 1 to 37:

Ex 7.2 Class 12 Maths Question 1.
2×1+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+2×2−−−−−−√
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+3×3)3
Solution:

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

Ex 7.2 Class 12 Maths Question 15.
x9−4×2
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−1×1+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−1×1−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−1×4)1+x8dx
Solution:

Choose the correct answer in exercises 38 and 39

Ex 7.2 Class 12 Maths Question 38.
∫10×9+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:

Move to Top → Class 12 Integrals Chapter 7

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:

Move to Top → Class 12 Integrals Chapter 7

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+4×2√
Solution:

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

Ex 7.4 Class 12 Maths Question 4.
19−25×2√
Solution:

Ex 7.4 Class 12 Maths Question 5.
3×1+2×4
Solution:

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

Ex 7.4 Class 12 Maths Question 7.
x−1×2−1√
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.
1×2+2x+2√
Solution:

Ex 7.4 Class 12 Maths Question 11.

19×2+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+12×2+x−3√
Solution:

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


Ex 7.4 Class 12 Maths Question 18.
5x−21+2x+3×2
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+2×2+2x+3√
Solution:


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


Ex 7.4 Class 12 Maths Question 23.
5x+3×2+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−4×2√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:

Move to Top → Class 12 Integrals Chapter 7

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.
1×2−9
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+5×3−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+1×2−1
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.
1×4−1
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:

Move to Top → Class 12 Integrals Chapter 7

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−1×1−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−1×2)
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(2×1+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:

Move to Top → Class 12 Integrals Chapter 7

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.
1−4×2−−−−−−√
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:

Move to Top → Class 12 Integrals Chapter 7

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:

Move to Top → Class 12 Integrals Chapter 7

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(4×3−5×2+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+35×2+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+3×2+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:

Move to Top → Class 12 Integrals Chapter 7

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:

Move to Top → Class 12 Integrals Chapter 7

NCERT Solutions for Class 2 Maths Integration Class 12 Ex 7.11

Move to Top → Class 12 Integrals Chapter 7

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:

Miscellaneous Exercise Class 11 Maths Question 7:

Solution:

Miscellaneous Exercise Class 11 Maths Question 8:

Solution:

Miscellaneous Exercise Class 11 Maths Question 9:

Solution:

Miscellaneous Exercise Class 11 Maths Question 10:

Solution:

Miscellaneous Exercise Class 11 Maths Question 11:

Solution:

Miscellaneous Exercise Class 11 Maths Question 12:

Solution:

Miscellaneous Exercise Class 11 Maths Question 13:

Solution:

Miscellaneous Exercise Class 11 Maths Question 14:

Solution:

Miscellaneous Exercise Class 11 Maths Question 15:

Solution:

Miscellaneous Exercise Class 11 Maths Question 16:

Solution:

Miscellaneous Exercise Class 11 Maths Question 17:

Solution:

Miscellaneous Exercise Class 11 Maths Question 18:

Solution:

Miscellaneous Exercise Class 11 Maths Question 20:

Solution:

Miscellaneous Exercise Class 11 Maths Question 21:

Solution:

Miscellaneous Exercise Class 11 Maths Question 22:

Solution:

Miscellaneous Exercise Class 11 Maths Question 23:

Solution:

Miscellaneous Exercise Class 11 Maths Question 24:

Solution:

Miscellaneous Exercise Class 11 Maths Question 25:

Solution:

Miscellaneous Exercise Class 11 Maths Question 26:

Solution:

Miscellaneous Exercise Class 11 Maths Question 27:

Solution:

Miscellaneous Exercise Class 11 Maths Question 28:

Solution:

Miscellaneous Exercise Class 11 Maths Question 29:

Solution:

Miscellaneous Exercise Class 11 Maths Question 30:

Miscellaneous Exercise Class 11 Maths Question 31:

Solution:

Miscellaneous Exercise Class 11 Maths Question 32:

Solution:

Miscellaneous Exercise Class 11 Maths Question 33:

Solution:

Miscellaneous Exercise Class 11 Maths Question 34:

Solution:

Question 35:

Solution:

Miscellaneous Exercise Class 11 Maths Question 36:

Solution:

Miscellaneous Exercise Class 11 Maths Question 37:

Solution:

Miscellaneous Exercise Class 11 Maths Question 38:

Solution:

Miscellaneous Exercise Class 11 Maths Question 39:

Solution:

Miscellaneous Exercise Class 11 Maths Question 40:

Solution:

Miscellaneous Exercise Class 11 Maths Question 41:

Solution:

Miscellaneous Exercise Class 11 Maths Question 42:

Solution:

Miscellaneous Exercise Class 11 Maths Question 43:

Solution:

Miscellaneous Exercise Class 11 Maths Question 44:

Solution:

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Integration Formulas