Find the best tutors and institutes for Class 12 Tuition
Q21:
Integrate the functions in Exercises 1 to 37:
21. $\tan^2(2x-3)$
Solution :
Step 1: Application of Trigonometric Identity
To integrate the function $\int \tan^2(2x-3) \, dx$, we utilize the fundamental trigonometric identity derived from the Pythagorean identity $\sec^2 \theta - \tan^2 \theta = 1$. Rearranging this, we obtain:
$\tan^2 \theta = \sec^2 \theta - 1$
Substituting $\theta = 2x - 3$, the integral becomes:
$\int \tan^2(2x-3) \, dx = \int (\sec^2(2x-3) - 1) \, dx$
Step 2: Decomposition and Integration
By the linearity property of integration [Per the Linearity of Integrals], we can split the expression into two distinct integrals:
$\int \sec^2(2x-3) \, dx - \int 1 \, dx$
Step 3: Evaluating the Integrals
For the first term, we apply the integration formula $\int \sec^2(u) \, du = \tan(u) + C$. Let $u = 2x - 3$. Then, $du = 2 \, dx$, or $dx = \frac{1}{2} \, du$ [By Substitution Rule].
$\int \sec^2(2x-3) \, dx = \frac{1}{2} \tan(2x-3)$
For the second term, we evaluate the integral of a constant:
$\int 1 \, dx = x + C_2$
Step 4: Synthesis of the Final Result
Combining the results from Step 3 and introducing the constant of integration $C$:
$\int \tan^2(2x-3) \, dx = \frac{1}{2} \tan(2x-3) - x + C$
Visual Representation of Function Transformation
Final Solution: \frac{1}{2} \tan(2x-3) - x + C
More Questions from Class 12 Mathematics Integrals EXERCISE 7.2
- Q1: Integrate the functions in Exercises 1 to 37: 1. $\frac{2x}{1+x^2}$
- Q2: Integrate the functions in Exercises 1 to 37: 2. $\frac{(\log x)^2}{x}$
- Q3: Integrate the functions in Exercises 1 to 37: 3. $\frac{1}{x + x \log x}$
- Q4: Integrate the functions in Exercises 1 to 37: 4. $\sin x \sin (\cos x)$
- Q5: Integrate the functions in Exercises 1 to 37: 5. $\sin (ax + b) \cos (ax + b)$
- Q6: Integrate the functions in Exercises 1 to 37: 6. $\sqrt{ax+b}$
- Q7: Integrate the functions in Exercises 1 to 37: 7. $x \sqrt{x+2}$
- Q8: Integrate the functions in Exercises 1 to 37: 8. $x \sqrt{1+2x^2}$
- Q9: Integrate the functions in Exercises 1 to 37: 9. $(4x+2)\sqrt{x^2+x+1}$
- Q10: Integrate the functions in Exercises 1 to 37: 10. $\frac{1}{x - \sqrt{x}}$
- Q11: Integrate the functions in Exercises 1 to 37: 11. $\frac{x}{\sqrt{x+4}}$, $x > 0$
- Q12: Integrate the functions in Exercises 1 to 37: 12. $(x^3 - 1)^{\frac{1}{3}} x^5$
- Q13: Integrate the functions in Exercises 1 to 37: 13. $\frac{x^2}{(2+3x^3)^3}$
- Q14: Integrate the functions in Exercises 1 to 37: 14. $\frac{1}{x(\log x)^m}$, $x > 0, m \neq 1$
- Q15: Integrate the functions in Exercises 1 to 37: 15. $\frac{x}{9-4x^2}$
- Q16: Integrate the functions in Exercises 1 to 37: 16. $e^{2x+3}$
- Q17: Integrate the functions in Exercises 1 to 37: 17. $\frac{x}{e^{x^2}}$
- Q18: Integrate the functions in Exercises 1 to 37: 18. $\frac{e^{\tan^{-1} x}}{1+x^2}$
- Q19: Integrate the functions in Exercises 1 to 37: 19. $\frac{e^{2x}-1}{e^{2x}+1}$
- Q20: Integrate the functions in Exercises 1 to 37: 20. $\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$
- Q22: Integrate the functions in Exercises 1 to 37: 22. $\sec^2(7-4x)$
- Q23: Integrate the functions in Exercises 1 to 37: 23. $\frac{\sin^{-1} x}{\sqrt{1-x^2}}$
- Q24: Integrate the functions in Exercises 1 to 37: 24. $\frac{2\cos x - 3\sin x}{6\cos x + 4\sin x}$
- Q25: Integrate the functions in Exercises 1 to 37: 25. $\frac{1}{\cos^2 x (1-\tan x)^2}$
- Q26: Integrate the functions in Exercises 1 to 37: 26. $\frac{\cos \sqrt{x}}{\sqrt{x}}$
- Q27: Integrate the functions in Exercises 1 to 37: 27. $\sqrt{\sin 2x} \cos 2x$
- Q28: Integrate the functions in Exercises 1 to 37: 28. $\frac{\cos x}{\sqrt{1+\sin x}}$
- Q29: Integrate the functions in Exercises 1 to 37: 29. $\cot x \log \sin x$
- Q30: Integrate the functions in Exercises 1 to 37: 30. $\frac{\sin x}{1+\cos x}$
- Q31: Integrate the functions in Exercises 1 to 37: 31. $\frac{\sin x}{(1+\cos x)^2}$
- Q32: Integrate the functions in Exercises 1 to 37: 32. $\frac{1}{1+\cot x}$
- Q33: Integrate the functions in Exercises 1 to 37: 33. $\frac{1}{1-\tan x}$
- Q34: Integrate the functions in Exercises 1 to 37: 34. $\frac{\sqrt{\tan x}}{\sin x \cos x}$
- Q35: Integrate the functions in Exercises 1 to 37: 35. $\frac{(1+\log x)^2}{x}$
- Q36: Integrate the functions in Exercises 1 to 37: 36. $\frac{(x+1)(x+\log x)^2}{x}$
- Q37: Integrate the functions in Exercises 1 to 37: 37. $\frac{x^3 \sin(\tan^{-1} x^4)}{1+x^8}$
- Q38: Choose the correct answer in Exercises 38 and 39. 38. $\int \frac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x} dx$ equals
- Q39: Choose the correct answer in Exercises 38 and 39. 39. $\int \frac{dx}{\sin^2 x \cos^2 x}$ equals
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.3
- 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
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Integrals
Top Tutor
Gharoli Gharoli, Delhi
16 yrs of
Exp
1 student
213 hrs of classes completed
600
per hour
Online Classes
Tutor's home
I bring with me 17 years of diverse experience in the pharmaceutical industry along with academic expertise, combining strong practical knowledge with a passion for teaching. I am dedicated to simplifying complex concepts and delivering them in an engaging manner, ensuring that students not only gain subject knowledge but also develop the skills, confidence, and potential required to achieve academic excellence and become industry-ready professionals.
Good tech and good behavior and good teaching style and full topic clear and all of best teacher and my side to best teacher.
Alamganj, Patna
10 yrs of
Exp
Online Classes
I am an experienced English language and literature faculty. I completed my masters in MA from Patna University in the year 2016 and currently pursuing a PhD in English Literature. I am working as a Speaking Examiner with Cambridge Assessment English, UK. I have 10 years of experience in teaching Spoken English, BCom, BCA, BSc, BTech and Class X-XII selected subjects for ISC/ICSE, State and CBSE Board students. I have taught students of different age groups and demography. My classes are interactive and learner-oriented. I follow various aspects of training with special Emphasis on Pronunciation.
Top Tutor
Rajarajeshwari Nagar, Bangalore
12 yrs of
Exp
51 students
1960 hrs of classes completed
500
per hour
Online Classes
Student's Home
I have been training students for 12th Board examinations for more than a decade now.
He is outstanding as a teacher, knowledgeable and dedicated. He has helped my daughter so much with his knowledge, that the girl who was just running away from accounts is getting more than 90 percent
Super Tutor
H C Sarani, Kolkata
18 yrs of
Exp
16 students
2068 hrs of classes completed
600
per hour
Online Classes
Tutor's home
Great
Sir is a dedicated and knowledgeable teacher who explains complex topics with clarity and patience. His classes are well-structured, engaging, and focused on practical understanding as well as theory. Overall, a highly effective and professional educator.
Top Tutor
Tri Nagar, Delhi
20 yrs of
Exp
4 students
32 hrs of classes completed
1000
per hour
Online Classes
I am Anshu Gupta. I was working at a reputed school in Delhi for 23 year before relocating to Singapore in 2021. I was teaching Physics to Grade 12 students. I have taught more than 3000 students in the last 24 years.I worked in Singapore as a freelance tutor for last 1.5 years I'm teaching Physics to the students of IB, IP, CBSE, CIE, ICSE, AP. I have a vast 24 year experience of teaching Physics to high school students I create individual based strategies for my students for best results I can create my own study material Good conceptual knowledge of my subject is my strength. I believe in creating a friendly and comfortable environment for free interaction. I have a lot of experience of online teaching ( 4 years ) So the online lessons are as effective as face to face lessons I'm teaching Physics to the students of IB(HL and SL), IP(H1 and H2), CBSE(11th and 12th), CIE(A and AS), ICSE, AP(Physics C). I am teaching 3 H2 and 3 IB students online at present
I craft this review as an offering of gratitude for the transformative influence she has on me. Please note that my review is completely transparent and true to my personal experience. I approached her with the request of covering Calculus from Class 12 RD Sharma (which extends to over 500 pages!). It's a challenge to find educators who truly prioritize depth and detail in their instruction and are willing to navigate the complexity that CBSE demands. However, under her guidance, we covered each exercise comprehensively, thoroughly and meticulously. Her teaching skills are truly top-notch and her sessions are sharply structured. Complex problems were masterfully explained. She is reservoir of knowledge, and always has all the answers to my doubts. Additionally, she keeps me engaged throughout the class, giving me questions to work through and think about independently. But it's not just her teaching that mesmerizes – it's her respect for punctuality that sets the tone for disciplined learning. Each class, every session, begins on time and makes it clear that she is a seasoned professional with kernels of experience to her name. All in all, if you're looking for an exceptional teacher, in fact the best on this platform, Deepa ma'am should be your only choice.
