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RD Sharma Class 12 Exercise 18.32 Indefinite Integrals Solutions Maths - Download PDF Free Online

RD Sharma Class 12 Exercise 18.32 Indefinite Integrals Solutions Maths - Download PDF Free Online

Edited By Kuldeep Maurya | Updated on Jan 24, 2022 10:54 AM IST

RD Sharma books are considered as some of the best materials to study for exams. Majority of CBSE schools and teachers follow this book for their lectures as well as the set of question papers. This is why it is a leading choice among students as it is informative and detailed. RD Sharma Solutions Indefinite Integrals Ex 18.32 material is prepared by a group of experts to help students get a better understanding of the subject. The Class 12 RD Sharma chapter 18 exercise 18.32 solution meets a fantastic essential of maths questions. It contains step-by-step solutions for each answer making it easier for students to understand.

RD Sharma Class 12 Solutions Chapter18 Indefinite Integrals - Other Exercise

Indefinite Integrals Excercise:18.32


Indefinite Integrals Excercise 18.32 Question 2

Answer : - 15log|2x+352x+3+5|+C
Hint :- Use substitution method to solve this integral.
Given :- 1(x1)2x+3dx
Sol : - Let I=1(x1)2x+3dx
Put 2x+3=t22dx=2tdtdx=tdt .then,
=1(t252)dt=2t25dt=2t2(52)dt
=2125log|t5t+5|+C(1x2a2dx=1/2alog|xax+a|+C)=15log|2x+352x+3+5|+C(t=2x+3) Ans.. .

Indefinite Integrals Excercise 18.32 Question 3

Answer : - 2x+2+23log|x+23x+2+3|+C
Hint :- Use substitution method to solve this integral.
Given :- x+1(x1)x+2dx
Sol : -
 Let I=x+1(x1)x+2dx=x1+2(x1)x+2dxI=(x1)+2(x1)x+2dxI={(x1)(x1)x+2+2(x1)x+2}dx
I=1x+2dx+2(x1)x+2dx
Now, I = I1 +I2 ........................ (I)
Where I1=1x+2dx and I2=2(x1)x+2dx
Therefore, I1=1x+2dx
Let x+2=udx=du Then,
I1=1u12du=u12du=u12+112+1+C(xndx=xn+1n+1+C)=u1212+C1=2x+2+C1 (II) 
And I2=2(x1)x+2dx
Let x+2=t2=>dx=2tdt. Then,
I2=2(t221)t22tdt=4tdt(t23)t(t22=x)=41(t2)32dt=423log|t3t+3|+C(1x2a2 dx=1/2alog|xax+a|+C)=23log|x+23x+2+3|+C2(t=x+2).III)

Put the value of Equ.(II) and (III) in (I) then,
I=2x+2+C1+23log|x+23x+2+3|+C2=2x+2++23log|x+23x+2+3|+C(C=C1+C2) Ans.. 

Indefinite Integrals Excercise 18.32 Question 4

Answer :23(x+2)322x+2+13log|x+23x+2+3|+C
Hint :- Use substitution method and special integration formula to solve this problem.
Given :- x2(x1)x+2dx
Sol :-
 let I=x2(x1)x+2dx=(x21)+1(x1)x+2dx=(x21)+1(x1)x+2dx={x21(x1)x+2+1(x1)x+2}dx={(x1)(x+1)(x1)x+2+1(x1)x+2}dx(a2b2=(a+b)(ab))
={x+1x+2+1(x1)x+2}dx={x+21x+2+1(x1)x+2}dx={(x+2)1x+2+1(x1)x+2}dx={(x+2)x+21x+2+1(x1)x+2}dx
={(x+2)1121x+2+1(x1)x+2}dx={x+21x+2+1(x1)x+2}dxI=x+2dx1x+2dx+1(x1)x+2 dx (I) 
Now,x+2dx
Put x+2=p=>dx=dp then
x+2dx=pdp=p12d=p12+112+1+C1(xndx=xn+1n+1+C)=p823/2=2/3(x+2)3/2+C1...(2)(p=x+2)
And, 1x+2dx
Put x+2=U=>dx=du, then
1x+2dx=1udu=u1/2du=u12+11/2+1+C2(xndx=xn+1n+1+C)=u12+11/2+1+C2=2u+C2=2x+2+C2..(III)(u=x+2)
Also, 1(x1)x+2dx
Put x+2=t2=>dx=2tdt, then
1(x1)x+2dx=1(t221)t22tdt=21(t23)ttdt=21t2(3)2 dt
=212.3log|t3t+3|+C3(1x2a2dx=12alog|xax+a|+C=13log|x+23x+2+3|+C3.(IV)
Put the values of equ.(II), (III) and (IV) in (I), then,
I=23(x+2)3/2+C1(2x+2+C2)+13log|x+23x+2+3|+C1I=23(x+2)3/22x+2+13log|x+23x+2+3|+C(C=C1C2+C3) Ans.. 

Indefinite Integrals Excercise 18.32 Question 5

Answer : - 2x+1+32log|x+12x+1+2|+C
Hint :- Use substitution method and special integration formula to solve this problem..
Given :- x(x3)x+1dx
Sol : - Let I=x(x3)x+1dx
=x3+3(x3)x+1dx=(x3)+3(x3)x+1dx={x3(x3)x+1+3(x3)x+1}dx={1x+1+3(x3)x+1}dx=1x+1dx+3(x3)x+1dx=1x+1dx+31(x3)x+1dx
Now,1x+1dx
Let x+1=p=>dx=dp. Then,
1x+1dx=1pdp=p1/2dp=p1/2+11/2+1+C1(xndx=xn+1n+1+C)=p1/21/2+C1=2x+1+C11(II)(p=x+1)

And, 1(x3)x+1dx

Put x+1=t2=>dx=2tdt. Then,
1(x3)x+1dx=1(t213)t22tdt(x=t21)=21(t24)ttdt=21(t2)(2)2 dt
=212.2log|t2t+2|+C2(1x2a2dx=12alog|xax+a|+C)=12log|x+12x+1+2|+C2. III )(t=x+1)
Put the values of equ. (II) and (III) in (I), Then ,
I=2x+1+C1+312log|x+12x+1+2|+C2I=2x+1+32log|x+12x+1+2|+C(C=C1+C2) Ans. 

Indefinite Integrals Excercise 18.32 Question 6

Answer : - 12tan1x12x122log|x+12xx+1+2x|+C
Hint :- Use substitution method and special integration formula.
Given :-1(x2+1)xdx
Sol : - Let I=1(x2+1)xdx
Put x=t2=>dx=2tdt. then,
I=1((t2)2+1)t22tdt=1(t4+1)t2tdt
=21(t4+1)dt=21t2t4+1t2dt (? dividing numerator and denominator by t2)

=21t2t4t2+1t2dt=21t2t2+1t2dt=1t2+1t2t2+1t2dt=1t2+11+1t2t2+1t2dt=(1+1t2)+(1t21)t2+1t2dt=(1+1t2)+(1t21)t2+1t2dt
=(1+1t2)t2+1t2dt(11t2)t2+1t2dt=(1+1t2)t2+1t2+22dt(11t2)t2+1t2+22dt=(1+1+t2)(t1t)2)+2dt(11t2)(t+1t)2)2dt (I) ((a+b)2=a2+2ab+b2(ab)2=a22ab+b2)
 Now, (1+1t2)(t1t)2+2)dt Put t 1/t=u=>1(1t2)dt=du(1+1t2)dt=du then (1+1t2)(t1t)2+2)dt=du(u)2+2)=du(u)2+(2)2)
=12tan1(u2)+C1(dxa2+x2=1atan1(xa)+C)=12tan1t1t2+C1(u=t1/t)=12tan1t21t+C1=12tan1t212t+C1... (II) 


And (11t2)(t+1t)22)dt

Put t+1/t=v=>(11/t2)dt=dy then,
(11t2)(t+1t)22)dt=dv(v)2+2)=dv(v)2+(2)2)=12.2log|v2v+2|+C(1x2a2dx=12alog|xax+a|+C)=122log|(t+1t2)(t+1t+2)|+C2
=12.2log|(t2+12t)(t2+1+2tt)|+C2=12.2log|(t2+12t)(t2+1+2t)|+C2..(III)
Put the values of equ. (II) and (III) in equ.(I), We get,
I=12tan1((t21)(2t)+C1(122log|(t2+12t)(t2+1+2t)|+C2)I=12tan1((t21)(2t)+C112.2log|(t2+12t)(t2+1+2t)|C2)I=12tan1((x1)(2x)12.2log|(x+12x)(x+1+2x)|+C)(t2=x and c=c1c2) Ans.. 

Indefinite Integrals Excercise 18.32 Question 7

Answer : - 12log|x+22(x+1)x+2+2(x+1)|+C
Hint :- Use substitution and special integration formula.
Given :- x(x2+2x+2)x+1dx
Sol : -
 Let I=x(x2+2x+2)x+1dx Put ,x+1=t2dx=2tdt then 
I=t21{(t21)2+2(t21)+2}t2.2 tdt (t21=x)=(t21)(t4+12t2+2t22+2)t2tdt=2t21(t4+1)dt
=2t21t2t4+1t2dt [Dividing num. And denom. By t2]
=2t221t2t4t2+1t2dt=2(11t2)t2+1t2dt
=2(11t2)t2+1t2+22dt=2(11t2)(t2+1t2+2.tt1)2dt=2(11t2)(t+1t)22 dt [(a+b)2=a2+2ab+b2]
 Again, put t+1t=u(11t2)dt=du then I=2duu2(2)2

=2122log|u2u+2|+c[1x2a2dx=12alog|xax+a|+c]=12log|t+1t2t+1t+2|+c[u=t+1t]=12log|t2+12ttt2+1+2tt|+c

=12log|t2+12tt2+1+2t|+c=12log|x+1+12x+1x+1+1+2x+1|+c[t=x+1 or t2=x+1]=12log|x+22(x+1)x+2+2(x+1)|+c

Indefinite Integrals Excercise 18.32 Question 8

Answer : -12log|(1x1+12)+(1x1+12)2+14|+C
Hint :- Use substitution and special integration formula.
Given :- 1(x1)x2+1dx
Sol : - Let I=1(x1)x2+1dx
Put, x1=1t
dx=1t2dt then,
I=11t(1t+1)2+1(1t2)dt=1t21t1t2+1+3t+1dt[(a+b)2=a2+b2+2ab]
=1t21t1+t2+2t+t2t2dt=1t211t1t2t2+2t+1dt=1t21t22(t2+t+12)dt
=121t2+2t12+(12)2(12)2+12dt=121(t+12)214+12dt[a2+b2+2ab=(a+b)2]
=121(t+12)2(124)dt=121(t+12)2(14)dt=121(t+12)2+14dt=121(t+12)2+(12)2dt
Put, t+12=u

dt=du than

I=121u2+(12)2duI=12log|u+u2+(12)2|+c[1x2+a2dx=log|x+x2+a2|+c]
I=12log|(t+12)+(t+12)2+14|+c(u=t+12)I=12log|(1x1+12)+(1x1+12)2+14|+c((x1)=1t)

Indefinite Integrals Excercise 18.32 Question 9

Answer : - log|(1x+112)+(1x+112)2+34|+C
Hint :- Use substitution method and special integration formula to solve this integration
Given :- 1(x+1)x2+x+1dx
Sol : - Let I=1(x+1)x2+x+1dx
Put, x+1=1t
dx=1t2dt then,
I=11t(1t1)2+(1t1)+1(1t2)dt(x=1t1)=1t21t1t2+12t+1t1+1dt(ab)2=a22ab+b2=1t21tt21t+1dt
=1t21t1t+t2t2dt=1t21t2t¯1t+t2dt=1t21t2t2t+1dt=1t22t212+(12)2(12)2+1dt

=1(t12)214+1dt=1(t12)2(144)dt

=1(t12)2(34)dt=1(t12)2+(34)dt=1(t12)2+(32)2dt
Put,t12=u
dt=du than,
I=1u2+(32)2du=log|u+u2+(32)2|+c[1x2+a2dx=log|x+x2+a2|+c]
=log|(t12)+(t12)2+34|+c(u=t12)=log|(1x+112)+(1x+112)2+34|+c[t=1x+1]

Indefinite Integrals Excercise 18.32 Question 10

Answer : - 122log|2x+x2+12xx2+1|+C
Hint :- Use substitution method and special integration formula.
Given :- 1(x21)x2+1dx
Sol : - Let I=1(x21)x2+1dx
Put, x=1t
dx=1t2dt then,
I=1(1t21)1t2+1(1t2)dt(x=1t)I=1(1t2t2)1+t2t21t2dt
I=1t21t2(1t2)1t,1+t2dtI=1t2(1t2),1t1+t2dtI=tdt(1t2)1+t2

 Put, 1+t2=u22tdt=2uduI=1{1(u21)]u2uduI=1(1u2+1)uuduI=12u2duI=1(2)2u2du

I=122log|2+u2u|+c[1a2x2dx=12alog|a+xax|+c]I=122log|2+1+t221+t2|+c[1+t2=u2 or u=1+t2]
I=122log|2+1+1x221+1x2|+cI=122log|2+x2+1x22x2+1x2|+c[t=1x]
I=122log|2+x2+1x2x2+1x|+cI=122log|2x+x2+1x2xx2+1x|+cI=122log|2x+x2+12xx2+1|+c



Indefinite Integrals Excercise 18.32 Question 11

Answer : - 13tan1(x2+13)+C
Hint :- Use substitution method and special integration formula..
Given :- x(x2+4)x2+1dx
Sol : - Let I=x(x2+4)x2+1dx
 Put x2+1=t22xdx=2tdtxdx=tdt than, I=1(t2+3)t2tdt(x2+1=t2x2=t21x2+4=t21+4x2+4=t2+3)
I=1(t2+3)ttdtI=1t2+3dtI=1t2+(3)2dt
I=13tan1(t2)+c[1a2+x2dx=1atan1(xa)+c]I=13tan1(x2+13)+cI=13tan1(x2+13)+c[t=x2+1]

Indefinite Integrals Excercise 18.32 Question 12

Answer : - 12tan1(1x22x2)+C
Hint :- Use substitution method and special integration formula..
Given :-1(x2+1)1x2dx
Sol : - Let I=1(x2+1)1x2dx
Put x=1tdx=1t2dt then,
I=1(1+1t2)11t2(1t2)dt=1t2(t2+1t2)t21t2dt=1t21t2(t2+1)1tt21dt
=t(t2+1)t21dt
Put, t21=u22tdt=2udutdt=udu than,
I=udu(u2+1+1)u2[t21=u2t2=u2+1]=udu(u2+2)u=duu2+(2)2
=12tan1(u2)+c[1a2+x2dx=1atan1(xa)+c]=12tan1(t212)+c[u2=t21u=t21]=12tan1(t212)+c
=12tan1(1x212)+c[x=1tt=1x]=12tan1(1x2x22)+c=12tan1(1x22x2)+c

Indefinite Integrals Excercise 18.32 Question 13

Answer : - 1233log|11x+3x21211x3x212|+C
Hint :- Use substitution method and special integration formula.
Given :- 1(2x2+3)x24dx
Sol : -
 Let I=1(2x2+3)x24dx let, x=1tdx=1t2dt then 
I=1(21t2+3)1t24(1t2dt)=1(2+3t2t2)14t2t21t2dt=1t21t2(3t2+2)14t21tdt=t(3t2+2)14t2dt
 Put  . 14t2=p24.2tdt=2pdptdt=pdp4
I=1{3(1p24)+2}p2pdp4[14t2=p21p2=4t2t2=1p24]
I=14133p2+84ppdpI=144113p2dpI=131(113p2)dp
I=131(113)2p2dpI=1312113log|113+p113p|+c[1a2x2dx=12alog|a+xax|

=12311log|113+14t211314t2|+c[p2=14t2p=14t2]=1233log|11+314t2311314t23|+c

=1233log|11+3(14t2)113(14t2)|+c=1233log|11+3(141x2)113(141x2)|+c[x=1tt=1 m]
=1233log|11+3x212x2113x212x2|+c=1233log|11+1x3x212111x3x212|+c
=1233log|11x+3x212x11x3x212x|+c=1233log|11x+3x21211x3x212|+c



Indefinite Integrals Excercise 18.32 Question 14

Answer : - 125log|x2+95x2+9+5|+C
Hint :- Use substitution method and special integration formula.
Given :- x(x2+4)x2+9dx
Sol : -
 Let I=x(x2+4)x2+9dx put, x2+9=u22xdx=2uduxdx=u du than I=udu(u29+4)u2
I=udu(u25)u[x2+9=u2x2=u29]I=uu25 duI=uu2(5)2 du
I=125log|u5u+5|+c[1x2a2dx=12alog|xax+a|+c]I=125log(x2+95x2+9+5)+C[x¨2+9=u2u=x2+9]


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RD Sharma Class 12th exercise 18.32 answers that are immediate and assessed. It further assists students with understanding formulae and keeping an eye on methods. RD Sharma class 12 chapter 18 exercise 18.32 has around 14 questions, which have subparts. Given below are some of the topics covered:

  • Some huge integrals close by theorems

  • A blend of sensible mathematical limits by using inadequate parts

  • Exactly when the denominator contains some reiterating straight factors, then the denominator contains unchangeable quadratic factors.

  • Blend of some interesting absurd arithmetical limits

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There are a total of 14 questions in RD Sharma Solutions Class 12 RD Sharma chapter 18 exercise 18.32.

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