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RD Sharma Class 12 Exercise 18.22 Indefinite integrals Solutions Maths - Download PDF Free Online

RD Sharma Class 12 Exercise 18.22 Indefinite integrals Solutions Maths - Download PDF Free Online

Edited By Kuldeep Maurya | Updated on Jan 24, 2022 12:26 PM IST

The Class 12 RD Sharma chapter 18 exercise 18.22 solution is the best solution for the chapter of Indefinite integrals for the students of class 12th. Students of any board can take these solutions for help if they want to score high in the exams. RD Sharma Solutions The RD Sharma class 12th exercise 18.22 is highly recommended for solving the complex chapters like Indefinite integrals, not only teachers but students as well prefer these solutions for thorough practice.

This Story also Contains
  1. RD Sharma Class 12 Solutions Chapter18 Indefinite Integrals - Other Exercise
  2. Indefinite Integrals Excercise:18.22
  3. Below are mentioned few benefits of the RD Sharma class 12 solution chapter 18 exercise 18.22 :-

RD Sharma Class 12 Solutions Chapter18 Indefinite Integrals - Other Exercise

Indefinite Integrals Excercise:18.22


Indefinite Integrals Exercise 18.22 Question 1.

Answer: 16tan1(3tanx2)+c
Given: 14cos2x+9sin2xdx
Hint: Use substitution method
Solution: 14cos2x+9sin2xdx
Dividing numerator and denominator by cos2x
=1cos2x4cos2xcos2x+9sin2xcos2xdx=sec2xdx4+9tan2x
Now,Let
tanx=tsec2xdx=dt (Differentiating w.r.t to x)
 So, dt4+9t2
=19dt49+t2
=19×32[tan1(t23)]+c [dtt2+a2=1atan1(ta)+c]
=16tan1(3tanx2)+c [tanx=t]

Indefinite Integrals Exercise 18.22 Question 2.

Answer: 125tan1(2tanx5)+c
Given:14sin2x+5cos2xdx
Hint: Divide the numerator and denominator by cos2x and then use substitution method
Solution:
14sin2x+5cos2xdx
Dividing the numerator and denominator by cos2x
1cos2x4sin2xcos2x+5cos2xcos2xdx
=sec2x4tan2x+5dx
Let t=tanxdt=sec2xdx (Differentiating w.r.t to x)
Now, 14t2+5dt
=141t2+54dt [dtt2+a2=1atan1(ta)+c]
=14×25tan1(t52)+c
=125tan1(2tanx5)+c

Indefinite Integrals Exercise 18.22 Question 3.

Answer: 23tan1(2tanx+13)+c
Given: 22+sin2xdx
Hint: You must know about substitution method
Solution: 22+sin2xdx
=22+2sinxcosxdx=11+sinxcosxdx
Dividing the numerator and denominator by cos2x
=sec2xsec2x+tanxdx=sec2x1+tan2x+tanxdx
Let
tanx=tsec2dx=dt (Differentiating w.r.t x)
Now, 11+t2+tdt
=1t2+2t12+(12)2(12)2+1dt
=1(t+12)2+(32)2dt

=132tan1(t+1232)+c [dtt2+a2=1atan1(ta)+c]

=23tan1(2t+13)+c=23tan1(2tanx+13)+c

Indefinite Integrals Exercise 18.22 Question 4.

Answer: 123log|1+3tanx13tanx|+c
Given: cosxcos3xdx
Hint: Use the formula of cos3x
Solution:
cosxcos3xdx
=cosx4cos3x3cosxdx
(cos3x=4cos3x3cosx)
=14cos2x3dx
Dividing numerator and denominator by cos2x
=sec2x43sec2xdx=sec2x43(1+tan2x)dx
(sec2x=1+tan2x)
=sec2x13tan2xdx
Let
tanx=tsec2xdx=dt (Differentiating w.r.t x)
=113t2dt=131(13)2t2dt
=13×12×113log|13+t13t|+c [dta2t2=12alog|a+tat|+c]
=16×3log|1+3t13t|+c
=123log|1+3tanx13tanx|+c

Indefinite Integrals Exercise 18.22 Question 5.

Answer: 12tan1(2tanx)+c
Given: 11+3sin2xdx
Hint: Use the formula of sec2x and then apply substitution method
Solution: 11+3sin2xdx
Divide numerator and denominator by cos2x
=1cos2x1cos2x+3sin2xcos2xdx
=sec2xsec2x+3tan2xdx
=sec2x1+tan2x+3tan2xdx
(sec2x=1+tan2x)
=sec2x1+4tan2xdx
Let
t=tanx (Differentiating w.r.t x)
dt=sec2xdx
=11+4t2dt
=141(12)2+t2dt
=14×112tan1(t12)+c [dtt2+a2=1atan1(ta)+c]
=12tan1(2tanx)+c

Indefinite Integrals Exercise 18.22 Question 6.

Answer: 115tan1(3tanx5)+c
Given: 13+2cos2xdx
Hint: Divide Numerator and Denominator by cos2x and then apply substitution method.
Solution: 13+2cos2xdx
On dividing Numerator and Denominator by cos2x
=sec2x3sec2x+2
=sec2x3(1+tan2x)+2dx
(sec2x=1+tan2x)
=sec2x5+3tan2xdx
Let
t=tanx (Differentiating w.r.t x)
dt=sec2xdx
Now,15+3t2dt
=131((53)2+t2)dt
=13×35tan1(t53)+c [dtt2+a2=1atan1(ta)+c]
=115tan1(3tanx5)+c

Indefinite Integrals Exercise 18.22 Question 7.

Answer: 15log|tanx22tanx+1|+c
Given: 1(sinx2cosx)(2sinx+cosx)dx
Hint: Apply substitution method and divide Numerator and Denominator by cos2x
Solution:
1(sinx2cosx)(2sinx+cosx)dx
=12sin2x+sinxcosx4sinxcosx2cos2xdx
=12sin2x3sinxcosx2cos2xdx
On dividing Numerator and Denominator by cos2x
=sec2x2tan2x3tanx2dx
Let
tanx=t (Differentiate w.r.t x)
sec2dx=dt
Now,12t23t2dt
=121t232t1dt
=121t22t(34)+(34)2(34)21dt
=121(t34)2(54)2dt
=12×12×54log|t3454t34+54|+c [dtt2a2=12alog|tat+a|+c]
=12×25log|2t42t+1|+c=15log|2tanx42tanx+1|+c

Indefinite Integrals Exercise 18.22 Question 8.

Answer: tan1(tan2x)+c
Given: sin2xsin4x+cos4xdx
Hint: Divide Numerator and denominator by cos4x
Solution:
sin2xsin4x+cos4xdx (sin2x=2sinxcosx)
=2sinxcosxsin4x+cos4xdx
On dividing Numerator and denominator by cos4x
=2tanxsec2xdxtan4x+1
Let
tan2x=t2tanxsec2xdx=dt (Differentiate w.r.t x)
Now,1t2+1dt
=tan1t+c [dtt2+a2=1atan1(ta)+c]
=tan1(tan2x)+c

Indefinite Integrals Exercise 18.22 Question 10.

Answer: 12log|tanxtanx+2|+c
Given: 1sin2x+sin2xdx
Hint: Use the formula of sin 2x and then apply substitution method
Solution: 1sin2x+sin2xdx
=1sin2x+2sinxcosxdx (sin2x=2sinxcosx)
On dividing numerator and denominator by cos2x, we get
=sec2xtan2x+2tanxdx
Let
t=tanx
dt=sec2xdx (Differentiate w.r.t x)
Now, 1t2+2tdt
=1t2+2t+(1)2(1)2dt=1(t+1)2(1)2dt[dtt2a2=12alog|tat+a|+c]
=12log|t+11t+1+1|+c
=12log|tt+2|+c=12log|tanxtanx+2|+c

Indefinite Integrals Exercise 18.22 Question 11.

Answer: 12tan1(2tanx)+c
Given: 1cos2x+3sin2xdx
Hint: Divide numerator and denominator by cos2x and then use substitution method
Solution:
1cos2x+3sin2xdx(cos2x=cos2xsin2x)=1cos2xsin2x+3sin2xdx=1cos2x+2sin2xdx
On dividing numerator and denominator by cos2x, we get
=sec2x1+2tan2xdx
Let
tanx=tsec2xdx=dt (Differentiate w.r.t x)
Now,11+2t2dt
=121(12)2+t2dt [dtt2+a2=1atan1(ta)+c]
=12×21tan1(t12)+c=12tan1(2tanx)+c


RD Sharma class 12 solution of Indefinite Integrals exercise 18.22 contains 11 questions, covered the following topics that are mentioned below:-

  • Integration of trigonometric and inverse trigonometric functions

  • Integration of exponential functions

Below are mentioned few benefits of the RD Sharma class 12 solution chapter 18 exercise 18.22 :-

  • The RD Sharma class 12th exercise 18.22 solution is crafted by great experts that are famous in the field of mathematics and thus it gives a vast proportion of knowledge to students in a way that no school can provide.

  • The syllabus of NCERT is quite vast and difficult, therefore students face trouble while solving questions from it and that is why RD Sharma class 12 chapter 18 exercise 18.22 is recommended for solved questions provided in it.

  • The most important benefit of downloading the RD Sharma class 12th exercise 18.22 from the Career360 website is that it is available free of cost for download on the website.

  • The RD Sharma solutions are used by teachers for lectures and also to provide homework so that makes it easy for students if they practice from the solution they can always stay ahead of the class.

  • RD Sharma class 12th exercise 18.22 gives a brief idea about the concepts and explains it in easy, moderate and tough level which will prepare you to solve any type of question asked in the exam.

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Frequently Asked Questions (FAQs)

1. What are indefinite integrals?

It is an integral expressed without limits, and so containing an arbitrary constant.

2. What is the general formula for indefinite integrals?

Either way, the general antiderivative is x+c and so 1dx = x+c, where c is a constant.

3. What is the process of finding an integral called?

The process of finding an integral is also called integration.

4. Is the RD Sharma solution available online?

Yes, you can download the RD Sharma solution from the official website of Careers360

5. How many questions are there in RD Sharma exercise 18.28?

There are a total of 11 questions in RD Sharma exercise 18.28 for helpful knowledge

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