Find the best tutors and institutes for Class 12 Tuition
Q20:
Integrate the functions in Exercises 1 to 22.
20. $\frac{(x-3)e^x}{(x-1)^3}$
Solution :
Step 1: Manipulating the Integrand
Given the integral: $I = \int \frac{(x-3)e^x}{(x-1)^3} \, dx$.
To simplify the integrand, we express the numerator $(x-3)$ in terms of $(x-1)$ to facilitate partial fraction decomposition relative to the denominator $(x-1)^3$.
Rewrite $(x-3)$ as $(x-1) - 2$.
Substituting this into the integral:
$I = \int \frac{((x-1) - 2)e^x}{(x-1)^3} \, dx$$I = \int \left( \frac{x-1}{(x-1)^3} - \frac{2}{(x-1)^3} \right) e^x \, dx$
$I = \int \left( \frac{1}{(x-1)^2} - \frac{2}{(x-1)^3} \right) e^x \, dx$ [By Algebraic Simplification]
Step 2: Applying the Integration Formula
We utilize the standard integration identity: $\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C$.
Let $f(x) = \frac{1}{(x-1)^2}$.
Differentiating $f(x)$ with respect to $x$ using the Power Rule and Chain Rule:
$f'(x) = \frac{d}{dx} (x-1)^{-2}$$f'(x) = -2(x-1)^{-3} \cdot \frac{d}{dx}(x-1)$ [Per the Chain Rule]
$f'(x) = \frac{-2}{(x-1)^3}$
Step 3: Verification of the Identity
Our integrand is now in the form $e^x [f(x) + f'(x)]$, where:
- $f(x) = \frac{1}{(x-1)^2}$
- $f'(x) = -\frac{2}{(x-1)^3}$
Substituting these back into the integral expression:
$I = \int e^x \left[ \frac{1}{(x-1)^2} + \left( -\frac{2}{(x-1)^3} \right) \right] \, dx$Step 4: Final Integration
Applying the formula $\int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C$:
$I = e^x \cdot \left( \frac{1}{(x-1)^2} \right) + C$Final Solution: $\frac{e^x}{(x-1)^2} + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.6
- Q1: Integrate the functions in Exercises 1 to 22. 1. $x \sin x$
- Q2: Integrate the functions in Exercises 1 to 22. 2. $x \sin 3x$
- Q3: Integrate the functions in Exercises 1 to 22. 3. $x^2 e^x$
- Q4: Integrate the functions in Exercises 1 to 22. 4. $x \log x$
- Q5: Integrate the functions in Exercises 1 to 22. 5. $x \log 2x$
- Q6: Integrate the functions in Exercises 1 to 22. 6. $x^2 \log x$
- Q7: Integrate the functions in Exercises 1 to 22. 7. $x \sin^{-1} x$
- Q8: Integrate the functions in Exercises 1 to 22. 8. $x \tan^{-1} x$
- Q9: Integrate the functions in Exercises 1 to 22. 9. $x \cos^{-1} x$
- Q10: Integrate the functions in Exercises 1 to 22. 10. $(\sin^{-1} x)^2$
- Q11: Integrate the functions in Exercises 1 to 22. 11. $\frac{x \cos^{-1} x}{\sqrt{1-x^2}}$
- Q12: Integrate the functions in Exercises 1 to 22. 12. $x \sec^2 x$
- Q13: Integrate the functions in Exercises 1 to 22. 13. $\tan^{-1} x$
- Q14: Integrate the functions in Exercises 1 to 22. 14. $x (\log x)^2$
- Q15: Integrate the functions in Exercises 1 to 22. 15. $(x^2+1) \log x$
- Q16: Integrate the functions in Exercises 1 to 22. 16. $e^x(\sin x + \cos x)$
- Q17: Integrate the functions in Exercises 1 to 22. 17. $\frac{x e^x}{(1+x)^2}$
- Q18: Integrate the functions in Exercises 1 to 22. 18. $e^x (\frac{1+\sin x}{1+\cos x})$
- Q19: Integrate the functions in Exercises 1 to 22. 19. $e^x (\frac{1}{x} - \frac{1}{x^2})$
- Q21: Integrate the functions in Exercises 1 to 22. 21. $e^{2x} \sin x$
- Q22: Integrate the functions in Exercises 1 to 22. 22. $\sin^{-1} (\frac{2x}{1+x^2})$
- Q23: Choose the correct answer in Exercises 23 and 24. 23. $\int x^2 e^{x^3} dx$ equals
- Q24: Choose the correct answer in Exercises 23 and 24. 24. $\int e^x \sec x (1 + \tan x) dx$ 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 7.5
- 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
Vesu, Surat
20 yrs of
Exp
14 students
2475 hrs of classes completed
500
per hour
Online Classes
I have experience teaching Mathematics to Class 12 students, focusing on building strong conceptual understanding and problem-solving skills. I teach key topics such as Calculus (limits, derivatives, and integrals), Matrices, Determinants, Probability, and Vectors. My teaching approach includes explaining concepts step-by-step, solving examples on the board, and giving practice questions similar to board exam patterns. I regularly conduct revision tests, doubt-clearing sessions, and discuss previous years’ question papers to help students prepare effectively for board examinations. I encourage students to understand the logic behind formulas rather than memorizing them. My goal is to make mathematics easier, improve students’ confidence, and help them achieve good results in Class 12 board exams.
A good teacher who relates to the child n guides them to perform better. Has clarity n mastery over the subject he teaches.
Top Tutor
Tollygunge, Kolkata
5 yrs of
Exp
3 students
38 hrs of classes completed
800
per hour
Online Classes
I have been giving tuitions to Class 12 students since October 2015. I was then in 2nd year of BSc Physics from the renowned 200 years old university, Presidency University. In total I taught 2 Class 12 students in Kolkata and 9 Class 12 students in Delhi. Currently I am teaching a Class 11 student from JNU Quarters Maths, Physics and Chemistry via Zoom app. I have whiteboard, colourful pens and other requirements to ease the teaching in a great way.
He has great ability to teach students by making intricate concept clear.he is also very honest.he gives full attention to me and never made me feel uncomfortable during the class.
Top Tutor
Banashankari, Bangalore
4 yrs of
Exp
5 students
271 hrs of classes completed
600
per hour
Online Classes
I am a commerce postgraduate from Pune University. Currently I am pursuing Ph.D in Commerce at one of the prestigious institutes for Commerce studies- Christ University, Bangalore. I have taught all the subjects of commerce to 11th & 12th Students since 4 years and BCom since 2 years. I have a good command on economics & accounts. I have a passion for teaching and have good understanding and practical exposure of these subjects. I also havd remotely worked as a Manager of Finance for a private firm in Maharashtra which makes me well versed in practical application of the subjects I teach.
Maam is very understanding. Explains concepts differently from school teaching making easy to learn and remember.
Super Tutor
Talwandi Sector-B, Kota
10 yrs of
Exp
15 students
891 hrs of classes completed
1850
per hour
Online Classes
Student's Home
Tutor's home
My teaching exp.11 year in most reputed coaching in india. Mentor of jee top rank air-12 yatis agrawal, 19, 28 many more in top 100 in jee adavnced. Many of students got 100%tile in jee mains along with 100% marks in board exams. I also provide complete relevent study material like worksheet, practice sheet , dpps , assignment etc. Daily quizz based on assignment. Periodically test papers topic wise, minor and major as per shedule. I have helped thousands of students overcome their fear of maths.
Great Teaching Expertise High standard. Experienced Teacher with excellent knowledge. Thank you for reaching my son.
Top Tutor
Lekha Nagar, Nashik
10 yrs of
Exp
2 students
23 hrs of classes completed
400
per hour
Online Classes
Tutor's home
Taught 6 years in Junior College for class12 ( Science and Commerce stream ) as a Math teacher.
