Find the best tutors and institutes for Class 12 Tuition
Q21:
Integrate the functions in Exercises 1 to 23.
21. $\frac{x^2+x+1}{(x+1)^2(x+2)}$
Solution :
Step 1: Partial Fraction Decomposition Setup
Given the rational function $f(x) = \frac{x^2+x+1}{(x+1)^2(x+2)}$, since the degree of the numerator (2) is less than the degree of the denominator (3), the function is a proper fraction. Per the theory of Partial Fractions, we decompose it as follows:
$\frac{x^2+x+1}{(x+1)^2(x+2)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+2}$ [By Partial Fraction Decomposition Theorem]
Step 2: Determining Constants A, B, and C
Equating the numerators by multiplying both sides by $(x+1)^2(x+2)$:
$x^2+x+1 = A(x+1)(x+2) + B(x+2) + C(x+1)^2$
- To find $B$, set $x = -1$: $(-1)^2 + (-1) + 1 = B(-1+2) \implies 1 = B(1) \implies \mathbf{B = 1}$
- To find $C$, set $x = -2$: $(-2)^2 + (-2) + 1 = C(-2+1)^2 \implies 4 - 2 + 1 = C(1) \implies \mathbf{C = 3}$
- To find $A$, equate the coefficients of $x^2$: $1 = A + C \implies 1 = A + 3 \implies \mathbf{A = -2}$
Step 3: Integration of Partial Fractions
The integral becomes:
$\int \frac{x^2+x+1}{(x+1)^2(x+2)} dx = \int \left( \frac{-2}{x+1} + \frac{1}{(x+1)^2} + \frac{3}{x+2} \right) dx$
Applying the linearity property of integration [$\int (af+bg) dx = a\int f dx + b\int g dx$]:
$I = -2 \int \frac{1}{x+1} dx + \int (x+1)^{-2} dx + 3 \int \frac{1}{x+2} dx$
Step 4: Final Evaluation
Executing the standard integration rules [$\int \frac{1}{x} dx = \ln|x| + C$ and $\int x^n dx = \frac{x^{n+1}}{n+1}$]:
$I = -2 \ln|x+1| + \frac{(x+1)^{-1}}{-1} + 3 \ln|x+2| + C$
$I = 3 \ln|x+2| - 2 \ln|x+1| - \frac{1}{x+1} + C$
Final Solution: $3 \ln|x+2| - 2 \ln|x+1| - \frac{1}{x+1} + C$
More Questions from Class 12 Mathematics Integrals Miscellaneous Exercise on Chapter 7
- Q1: Integrate the functions in Exercises 1 to 23. 1. $\frac{1}{x-x^3}$
- Q2: Integrate the functions in Exercises 1 to 23. 2. $\frac{1}{\sqrt{x+a} + \sqrt{x+b}}$
- Q3: Integrate the functions in Exercises 1 to 23. 3. $\frac{1}{x\sqrt{ax-x^2}}$ [Hint: Put $x = \frac{a}{t}$]
- Q4: Integrate the functions in Exercises 1 to 23. 4. $\frac{1}{x^2(x^4+1)^{\frac{3}{4}}}$
- Q5: Integrate the functions in Exercises 1 to 23. 5. $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}$ [Hint: $\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}} = \frac{1}{x^{\frac{1}{3}}(1+x^{\frac{1}{6}})}$, put $x=t^6$]
- Q6: Integrate the functions in Exercises 1 to 23. 6. $\frac{5x}{(x+1)(x^2+9)}$
- Q7: Integrate the functions in Exercises 1 to 23. 7. $\frac{\sin x}{\sin(x-a)}$
- Q8: Integrate the functions in Exercises 1 to 23. 8. $\frac{e^{5\log x} - e^{4\log x}}{e^{3\log x} - e^{2\log x}}$
- Q9: Integrate the functions in Exercises 1 to 23. 9. $\frac{\cos x}{\sqrt{4-\sin^2 x}}$
- Q10: Integrate the functions in Exercises 1 to 23. 10. $\frac{\sin^8 x - \cos^8 x}{1 - 2\sin^2 x \cos^2 x}$
- Q11: Integrate the functions in Exercises 1 to 23. 11. $\frac{1}{\cos(x+a)\cos(x+b)}$
- Q12: Integrate the functions in Exercises 1 to 23. 12. $\frac{x^3}{\sqrt{1-x^8}}$
- Q13: Integrate the functions in Exercises 1 to 23. 13. $\frac{e^x}{(1+e^x)(2+e^x)}$
- Q14: Integrate the functions in Exercises 1 to 23. 14. $\frac{1}{(x^2+1)(x^2+4)}$
- Q15: Integrate the functions in Exercises 1 to 23. 15. $\cos^3 x e^{\log \sin x}$
- Q16: Integrate the functions in Exercises 1 to 23. 16. $e^{3\log x} (x^4+1)^{-1}$
- Q17: Integrate the functions in Exercises 1 to 23. 17. $f'(ax+b)[f(ax+b)]^n$
- Q18: Integrate the functions in Exercises 1 to 23. 18. $\frac{1}{\sqrt{\sin^3 x \sin(x+\alpha)}}$
- Q19: Integrate the functions in Exercises 1 to 23. 19. $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$
- Q20: Integrate the functions in Exercises 1 to 23. 20. $\frac{2+\sin 2x}{1+\cos 2x} e^x$
- Q22: Integrate the functions in Exercises 1 to 23. 22. $\tan^{-1} \sqrt{\frac{1-x}{1+x}}$
- Q23: Integrate the functions in Exercises 1 to 23. 23. $\frac{\sqrt{x^2+1}[\log(x^2+1)-2\log x]}{x^4}$
- Q24: Evaluate the definite integrals in Exercises 24 to 31. 24. $\int_{\frac{\pi}{2}}^{\pi} e^x (\frac{1-\sin x}{1-\cos x}) dx$
- Q25: Evaluate the definite integrals in Exercises 24 to 31. 25. $\int_{0}^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} dx$
- Q26: Evaluate the definite integrals in Exercises 24 to 31. 26. $\int_{0}^{\frac{\pi}{2}} \frac{\cos^2 x dx}{\cos^2 x + 4\sin^2 x}$
- Q27: Evaluate the definite integrals in Exercises 24 to 31. 27. $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx$
- Q28: Evaluate the definite integrals in Exercises 24 to 31. 28. $\int_{0}^{1} \frac{dx}{\sqrt{1+x}-\sqrt{x}}$
- Q29: Evaluate the definite integrals in Exercises 24 to 31. 29. $\int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9+16\sin 2x} dx$
- Q30: Evaluate the definite integrals in Exercises 24 to 31. 30. $\int_{0}^{\frac{\pi}{2}} \sin 2x \tan^{-1}(\sin x) dx$
- Q31: Evaluate the definite integrals in Exercises 24 to 31. 31. $\int_{1}^{4} [|x-1|+|x-2|+|x-3|] dx$
- Q32: Prove the following (Exercises 32 to 37) 32. $\int_{1}^{3} \frac{dx}{x^2(x+1)} = \frac{2}{3} + \log \frac{2}{3}$
- Q33: Prove the following (Exercises 32 to 37) 33. $\int_{0}^{1} x e^x dx = 1$
- Q34: Prove the following (Exercises 32 to 37) 34. $\int_{-1}^{1} x^{17} \cos^4 x dx = 0$
- Q35: Prove the following (Exercises 32 to 37) 35. $\int_{0}^{\frac{\pi}{2}} \sin^3 x dx = \frac{2}{3}$
- Q36: Prove the following (Exercises 32 to 37) 36. $\int_{0}^{\frac{\pi}{4}} 2\tan^3 x dx = 1 - \log 2$
- Q37: Prove the following (Exercises 32 to 37) 37. $\int_{0}^{1} \sin^{-1} x dx = \frac{\pi}{2} - 1$
- Q38: Choose the correct answers in Exercises 38 to 40 38. $\int \frac{dx}{e^x + e^{-x}}$ is equal to
- Q39: Choose the correct answers in Exercises 38 to 40 39. $\int \frac{\cos 2x}{(\sin x + \cos x)^2} dx$ is equal to
- Q40: Choose the correct answers in Exercises 38 to 40 40. If $f(a+b-x) = f(x)$, then $\int_{a}^{b} x f(x) dx$ is equal to
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Integrals
Top Tutor
Karolan Ka Barh, Jaipur
10 yrs of
Exp
2 students
73 hrs of classes completed
700
per hour
Online Classes
Tutor's home
With 12 years of dedicated teaching experience, I have consistently aimed to deliver high-quality education in the field of Commerce and related subjects. Holding a B.Com, M.Com, and B.Ed, along with being UGC NET qualified, I have taught at various academic levels, ensuring conceptual clarity and practical understanding for students. My teaching approach emphasizes student engagement, real-life applications of theoretical knowledge, and consistent academic growth. I have successfully prepared students for board exams and competitive assessments, fostering both academic excellence and confidence. Throughout my career, I have remained committed to continuous learning and adapting to modern teaching methodologies.
Make complex topics easy to grasp and encourage the students and help in learning with such patient.
Top Tutor
CUC, Gachibowli, Hyderabad
5 yrs of
Exp
2 students
3 hrs of classes completed
375
per hour
Online Classes
I am avid learner of Science and have helped several students improve their understanding and hence academic performances.Currently,I am doing my Ph.d. in Mathematics from IISER Bhopal. I have done my Bachelors in Mathematics from IGNOU. I have also completed M.Sc. in Pure Mathematics from University of Hyderabad in 2024. I have successfully managed to clear GATE, NET JRF in Mathematical science..To spread my love for Mathematics and Physics I have been teaching Mathematics and Physics for the past 6 years (std 9-12, B.Sc and M.Sc). So ,If you want to understand the concepts of mathematics and physics, I am here to help you with that. I also provide Customized notes and exercise to focus on the area of improvement .Classes are recorded so that you can go through them anytime and refresh/ revise the topics. I have all the necessary tools for online classes. So, hope to see you in Demo.
It was a very nice experience. He went very in-depth and clear. It was very useful. I studied maths for CBSE board.
Super Tutor
Besant Nagar, Chennai
20 yrs of
Exp
23 students
895 hrs of classes completed
750
per hour
Online Classes
am geetha, from Chennai, a very dedicated and sincere professional from my end and I ensure that it is inculcated and reciprocated by the students as well. I have been in the teaching field since 2005 after my master's. I have tutored more 350 students in various batches and have boosted their confidence in Math, making the subject enjoyable for them. All of my past students have passed in flying colors and are flourishing in their professional careers now My class will be focused on strengthening the basic concepts and foundation that is essential to break down complex problems by using simple methods. I focus on clearing all the doubts step by step and conducting test papers after the completion of every lesson which will make it easier for the students to crack their school quarterly, half-yearly, and final exams with ease. Thank you, best regards from geetha Kumar.
Geetha Ma'am explains well and is ready to clear doubts. So far the experience has been very good and helpful to us.
Top Tutor
Vijay Nagar Colony Sector 9, Ghaziabad
7 yrs of
Exp
55 students
1067 hrs of classes completed
400
per hour
Online Classes
Founder of knowledge park classes. I am a graduate in corporate affairs and administration and CS Final too. I have experience of more than 10 years. Our results have been extra ordinary and one of our student from DPS greater Noida scored 100 marks in accounts in 2021-22 session. Over all all our students have been scoring 95 plus consistently from many years. I have successfully taught students of Amity, DPS, Ryan, gurukul among many others. I teach: Company account Financial management Business studies Mathematics Economics Taxation And all other subjects of Bcom,mcom,CA/CS/CMA.
It was an amazing experience learning with sir .He is absolutely amazing teacher he teaches everything in detail very nicely.
Top Tutor
Ramanthapur, Hyderabad
20 yrs of
Exp
28 students
94 hrs of classes completed
500
per hour
Online Classes
Student's Home
Tutor's home
I deal students according to their demand. I teach Accounts of state CBSE ISC all syllabus. While teaching I make sure students are understanding perfectly. This is possible when students are pretty confident in solving sums as many as possible for which they must get clear picture of topic. Concept should be cristle clear. I am able to deal any kind of students.
Sir cleared all the doubts and never had any problem adjusting with changing timings. Sir taught really well and made me perfectly ready for boards ensuring 95+
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Integrals Miscellaneous Exercise on Chapter 7 worksheets
Download Now
