Find the best tutors and institutes for Class 12 Tuition
Q11:
Integrate the rational functions in Exercises 1 to 21.
11. $\frac{5x}{(x+1)(x^2-4)}$
Solution :
Step 1: Decomposition of the Rational Function
To integrate $\int \frac{5x}{(x+1)(x^2-4)} dx$, we first factorize the quadratic term in the denominator. Recall the algebraic identity $a^2 - b^2 = (a-b)(a+b)$.
$x^2 - 4 = (x-2)(x+2)$
Therefore, the integrand becomes:
$\frac{5x}{(x+1)(x-2)(x+2)}$
[By Partial Fraction Decomposition theorem, since all factors are linear and distinct, we assume]:
$\frac{5x}{(x+1)(x-2)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{x+2}$
Step 2: Determining the Constants A, B, and C
Multiply both sides by $(x+1)(x-2)(x+2)$ to obtain the identity:
$5x = A(x-2)(x+2) + B(x+1)(x+2) + C(x+1)(x-2)$
$5x = A(x^2 - 4) + B(x^2 + 3x + 2) + C(x^2 - x - 2)$
- To find $A$, set $x = -1$: $5(-1) = A(-1-2)(-1+2) \implies -5 = A(-3)(1) \implies A = \frac{5}{3}$
- To find $B$, set $x = 2$: $5(2) = B(2+1)(2+2) \implies 10 = B(3)(4) \implies 10 = 12B \implies B = \frac{10}{12} = \frac{5}{6}$
- To find $C$, set $x = -2$: $5(-2) = C(-2+1)(-2-2) \implies -10 = C(-1)(-4) \implies -10 = 4C \implies C = -\frac{10}{4} = -\frac{5}{2}$
Step 3: Integration of the Partial Fractions
The integral can now be expressed as the sum of three elementary integrals:
$\int \frac{5x}{(x+1)(x^2-4)} dx = \frac{5}{3} \int \frac{1}{x+1} dx + \frac{5}{6} \int \frac{1}{x-2} dx - \frac{5}{2} \int \frac{1}{x+2} dx$
[Applying the standard integral rule $\int \frac{1}{x} dx = \ln|x| + C$]:
$= \frac{5}{3} \ln|x+1| + \frac{5}{6} \ln|x-2| - \frac{5}{2} \ln|x+2| + C$
Visual Representation of Partial Fraction Components
Step 4: Simplification
To provide a more compact form, we observe the coefficients $\frac{5}{3}, \frac{5}{6}, -\frac{5}{2}$. Finding a common denominator ($6$):
$= \frac{10}{6} \ln|x+1| + \frac{5}{6} \ln|x-2| - \frac{15}{6} \ln|x+2| + C$
$= \frac{5}{6} [2\ln|x+1| + \ln|x-2| - 3\ln|x+2|] + C$
[Using logarithmic power rule $n\ln(a) = \ln(a^n)$ and quotient rule $\ln(a) - \ln(b) = \ln(a/b)$]:
$= \frac{5}{6} \ln \left| \frac{(x+1)^2(x-2)}{(x+2)^3} \right| + C$
Final Solution: \frac{5}{3} \ln|x+1| + \frac{5}{6} \ln|x-2| - \frac{5}{2} \ln|x+2| + C
More Questions from Class 12 Mathematics Integrals EXERCISE 7.5
- Q1: Integrate the rational functions in Exercises 1 to 21. 1. $\frac{x}{(x+1)(x+2)}$
- Q2: Integrate the rational functions in Exercises 1 to 21. 2. $\frac{1}{x^2-9}$
- Q3: Integrate the rational functions in Exercises 1 to 21. 3. $\frac{3x-1}{(x-1)(x-2)(x-3)}$
- Q4: Integrate the rational functions in Exercises 1 to 21. 4. $\frac{x}{(x-1)(x-2)(x-3)}$
- Q5: Integrate the rational functions in Exercises 1 to 21. 5. $\frac{2x}{x^2+3x+2}$
- Q6: Integrate the rational functions in Exercises 1 to 21. 6. $\frac{1-x^2}{x(1-2x)}$
- Q7: Integrate the rational functions in Exercises 1 to 21. 7. $\frac{x}{(x^2+1)(x-1)}$
- Q8: Integrate the rational functions in Exercises 1 to 21. 8. $\frac{x}{(x-1)^2(x+2)}$
- Q9: Integrate the rational functions in Exercises 1 to 21. 9. $\frac{3x+5}{x^3-x^2-x+1}$
- Q10: Integrate the rational functions in Exercises 1 to 21. 10. $\frac{2x-3}{(x^2-1)(2x+3)}$
- Q12: Integrate the rational functions in Exercises 1 to 21. 12. $\frac{x^3+x+1}{x^2-1}$
- Q13: Integrate the rational functions in Exercises 1 to 21. 13. $\frac{2}{(1-x)(1+x^2)}$
- Q14: Integrate the rational functions in Exercises 1 to 21. 14. $\frac{3x-1}{(x+2)^2}$
- Q15: Integrate the rational functions in Exercises 1 to 21. 15. $\frac{1}{x^4-1}$
- Q16: Integrate the rational functions in Exercises 1 to 21. 16. $\frac{1}{x(x^n+1)}$ [Hint: multiply numerator and denominator by $x^{n-1}$ and put $x^n = t$]
- Q17: Integrate the rational functions in Exercises 1 to 21. 17. $\frac{\cos x}{(1-\sin x)(2-\sin x)}$ [Hint : Put $\sin x = t$]
- Q18: Integrate the rational functions in Exercises 1 to 21. 18. $\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$
- Q19: Integrate the rational functions in Exercises 1 to 21. 19. $\frac{2x}{(x^2+1)(x^2+3)}$
- Q20: Integrate the rational functions in Exercises 1 to 21. 20. $\frac{1}{x(x^4-1)}$
- Q21: Integrate the rational functions in Exercises 1 to 21. 21. $\frac{1}{e^x-1}$ [Hint : Put $e^x=t$]
- Q22: Choose the correct answer in each of the Exercises 22 and 23. 22. $\int \frac{x dx}{(x-1)(x-2)}$ equals
- Q23: Choose the correct answer in each of the Exercises 22 and 23. 23. $\int \frac{dx}{x(x^2+1)}$ equals
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 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Integrals
Top Tutor
Narasapur, Gadag
10 yrs of
Exp
2 students
64 hrs of classes completed
400
per hour
Online Classes
I specialize in breaking down complex mathematical derivations, interpreting complex velocity-time or field graphs, and explaining the physical significance behind equations.Step-by-step algorithmic breakdown of complex integrals, visualizing 3D space geometries, and teaching the underlying logic of proofs.I break down intimidating, high-weightage topics into digestible sub-concepts. For example, before diving into complex Electromagnetic Induction problems, I ensure the student thoroughly grips Lenz's Law and magnetic flux concepts.
Do not underestimate him for being a small tutor; he is amazing! He is the most patient, steady, and supportive physics tutor you will ever find. He is literally the best, never once hesitating to explain something again and again. He is perfect and will do everything for you to have the best experience in class, and he is the best for conceptual clarity. No matter how many doubts you ask, he will never run out of explanations.
Top Tutor
Electronic City, Bangalore
11 yrs of
Exp
5 students
185 hrs of classes completed
500
per hour
Online Classes
Well experienced in physics and chemistry of XI , XII standard in both ICSE and CBSE curricular. Taken part time classes in college for ISC syllabus. Strong hold on Mechanics and Organic compounds and thermodynamics.
He explains the subject like the things really happening in front of eyes. The magnetism, optics, waves, Current etc. topics are now crystal clear. You get much more than the book from his lectures. His notes are so detailed that any layman can read and understand.
Top Tutor
Govind Puri, Delhi
4 yrs of
Exp
3 students
16 hrs of classes completed
900
per hour
Online Classes
I have teaching Biology from past 4 years with excellent scores. I have taught CBSE, IGCSE, IB boards of class 11th, 12th, and 9th 10th. I try to go deep into concepts and teach the students the core information which helps them understand Biology like nothing. I have had students scoring good at NEET, Boards and entrances with my teaching. See you in class.
Really liked the way she taught. Learnt a whole lot with very good understanding now. Looking forward for more.
Top Tutor
Aganampudi, Visakhapatnam
7 yrs of
Exp
2 students
65 hrs of classes completed
500
per hour
Online Classes
Student's Home
Since from last seven years i am doing for 10, 11 and 12 th class chemistry students and i am efficient in doubt solving of students.
Very nice explanation of each chapter without any doubts and easy to understand all concepts and easy to solve questions.
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.
