Find the best tutors and institutes for Class 12 Tuition
Q5:
Integrate the functions in Exercises 1 to 23.
5. $\frac{3x}{1+2x^4}$
Solution :
Step 1: Analyzing the Integrand
The objective is to evaluate the integral:
$I = \int \frac{3x}{1 + 2x^4} \, dx$
Observe that the numerator contains $x$, which suggests a substitution involving the derivative of $x^2$. We rewrite the denominator to express $x^4$ as a square: $2x^4 = (\sqrt{2}x^2)^2$.
Step 2: Substitution Method
Let $u = \sqrt{2}x^2$. [By the Method of Substitution]
Differentiating with respect to $x$:
$\frac{du}{dx} = 2\sqrt{2}x \implies dx = \frac{du}{2\sqrt{2}x}$
Step 3: Transforming the Integral
Substitute $u$ and $dx$ into the original integral:
$I = \int \frac{3x}{1 + u^2} \cdot \frac{du}{2\sqrt{2}x}$
Canceling $x$ from the numerator and denominator:
$I = \frac{3}{2\sqrt{2}} \int \frac{1}{1 + u^2} \, du$
Step 4: Integration Using Standard Formula
Apply the standard integral identity: $\int \frac{1}{1+t^2} \, dt = \tan^{-1}(t) + C$. [Per the Fundamental Theorem of Calculus]
$I = \frac{3}{2\sqrt{2}} \tan^{-1}(u) + C$
Step 5: Back-Substitution
Substitute $u = \sqrt{2}x^2$ back into the expression:
$I = \frac{3}{2\sqrt{2}} \tan^{-1}(\sqrt{2}x^2) + C$
Step 6: Rationalizing the Coefficient (Optional but Standard)
Multiply the numerator and denominator of the coefficient by $\sqrt{2}$:
$\frac{3}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{4}$
$I = \frac{3\sqrt{2}}{4} \tan^{-1}(\sqrt{2}x^2) + C$
Visual Representation of the Integrand Function
Final Solution: $\frac{3}{2\sqrt{2}} \tan^{-1}(\sqrt{2}x^2) + C$ or $\frac{3\sqrt{2}}{4} \tan^{-1}(\sqrt{2}x^2) + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.4
- Q1: Integrate the functions in Exercises 1 to 23. 1. $\frac{3x^2}{x^6+1}$
- Q2: Integrate the functions in Exercises 1 to 23. 2. $\frac{1}{\sqrt{1+4x^2}}$
- Q3: Integrate the functions in Exercises 1 to 23. 3. $\frac{1}{\sqrt{(2-x)^2+1}}$
- Q4: Integrate the functions in Exercises 1 to 23. 4. $\frac{1}{\sqrt{9-25x^2}}$
- Q6: Integrate the functions in Exercises 1 to 23. 6. $\frac{x^2}{1-x^6}$
- Q7: Integrate the functions in Exercises 1 to 23. 7. $\frac{x-1}{\sqrt{x^2-1}}$
- Q8: Integrate the functions in Exercises 1 to 23. 8. $\frac{x^2}{\sqrt{x^6+a^6}}$
- Q9: Integrate the functions in Exercises 1 to 23. 9. $\frac{\sec^2 x}{\sqrt{\tan^2 x + 4}}$
- Q10: Integrate the functions in Exercises 1 to 23. 10. $\frac{1}{\sqrt{x^2+2x+2}}$
- Q11: Integrate the functions in Exercises 1 to 23. 11. $\frac{1}{9x^2+6x+5}$
- Q12: Integrate the functions in Exercises 1 to 23. 12. $\frac{1}{\sqrt{7-6x-x^2}}$
- Q13: Integrate the functions in Exercises 1 to 23. 13. $\frac{1}{\sqrt{(x-1)(x-2)}}$
- Q14: Integrate the functions in Exercises 1 to 23. 14. $\frac{1}{\sqrt{8+3x-x^2}}$
- Q15: Integrate the functions in Exercises 1 to 23. 15. $\frac{1}{\sqrt{(x-a)(x-b)}}$
- Q16: Integrate the functions in Exercises 1 to 23. 16. $\frac{4x+1}{\sqrt{2x^2+x-3}}$
- Q17: Integrate the functions in Exercises 1 to 23. 17. $\frac{x+2}{\sqrt{x^2-1}}$
- Q18: Integrate the functions in Exercises 1 to 23. 18. $\frac{5x-2}{1+2x+3x^2}$
- Q19: Integrate the functions in Exercises 1 to 23. 19. $\frac{6x+7}{\sqrt{(x-5)(x-4)}}$
- Q20: Integrate the functions in Exercises 1 to 23. 20. $\frac{x+2}{\sqrt{4x-x^2}}$
- Q21: Integrate the functions in Exercises 1 to 23. 21. $\frac{x+2}{\sqrt{x^2+2x+3}}$
- Q22: Integrate the functions in Exercises 1 to 23. 22. $\frac{x+3}{x^2-2x-5}$
- Q23: Integrate the functions in Exercises 1 to 23. 23. $\frac{5x+3}{\sqrt{x^2+4x+10}}$
- Q24: Choose the correct answer in Exercises 24 and 25. 24. $\int \frac{dx}{x^2+2x+2}$ equals
- Q25: Choose the correct answer in Exercises 24 and 25. 25. $\int \frac{dx}{\sqrt{9x-4x^2}}$ 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 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Integrals
Super Tutor
Prem Nagar, Hisar
20 yrs of
Exp
47 students
10096 hrs of classes completed
800
per hour
Online Classes
Student's Home
I am very expert in teaching the basics with simple examples and make to understand the concepts very simple way. It will helpful for the students to understand and apply at a given situation especially application-based and high thinking problems. In fact, I will not teach the student but help to learn the subject in a more independent way and after understanding the level of comfort of the student.
We want to express our deepest gratitude to Sir for his incredible guidance in helping our daughter succeed. He is a highly experienced and knowledgeable educator who truly understands how to connect with students. When our daughter was first introduced to Accountancy in 11th grade, she found the new subject quite challenging and began losing interest. However, thanks to Sir’s consistent support and patience, her performance and confidence improved tremendously. He is deeply sincere, helpful, and always approachable when a student encounters difficulties. Any student would be incredibly fortunate to have him as a mentor, and we highly recommend his tutoring to anyone looking for excellent guidance in Accountancy.
Super Tutor
Katira Maharaja Hata, Arrah
4 yrs of
Exp
16 students
5824 hrs of classes completed
600
per hour
Online Classes
Tutor's home
I have successfully taught hundreds of Class 12 students across multiple boards, including CBSE, ICSE, IGCSE, International Curricula, and State Boards. My expertise spans Mathematics & Physics for students from science, commerce, and arts streams, with a strong focus on clarity and conceptual understanding. Alongside my individual teaching, I manage a coaching institute dedicated to preparing students from Class 10 to 12 and guiding IIT JEE aspirants with personalized strategies for success. Please note: The fees displayed on my profile apply to one-on-one personalized sessions. For details on group or batch classes, please refer to my monthly fee structure or contact me directly.
Shiwam is a good guide for my son. I love the tips and tricks he teaches in Maths so that it becomes easy for my son.
Top Tutor
New Bowenpally, Hyderabad
20 yrs of
Exp
27 students
440 hrs of classes completed
800
per hour
Online Classes
Tutor's home
every year 100s of my students pass out in chemistry more the 95% in CBSE, ISE and other state board exams
Vinod Sir is an exceptional faculty for IIT coaching. He explains complex concepts like Thermodynamics in a very simple, clear, and easy-to-understand manner. His teaching style makes even difficult topics interesting and accessible to every student. His strong conceptual approach and ability to simplify challenging problems are truly commendable. I highly recommend Vinod Sir to anyone preparing for IIT/JEE.
Super Tutor
Panvel, Mumbai
12 yrs of
Exp
21 students
3291 hrs of classes completed
500
per hour
Online Classes
I provide private tuition for Physics and Mathematics for Class XI and XII (All Boards) Physics for Engineering Aspirants Physics for Medical Aspirants Class IX and X Mathematics and Science.
I am grateful that my son have been taught by you. Your dedication, knowledge, and passion for teaching make every class engaging and productive. You explain complex concepts in a simple and understandable manner. Thank you Sir .
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.
