Find the best tutors and institutes for Class 12 Tuition
Q9:
Find the following integrals 9. $\int (2x^2 + e^x) dx$
Find the following integrals 9. $\int (2x^2 + e^x) dx$
Solution :
Step 1: Application of the Linearity Property of Integrals
The integral of a sum of two functions is equal to the sum of their individual integrals [Per the Linearity Property of Indefinite Integrals]:
$\int (2x^2 + e^x) \, dx = \int 2x^2 \, dx + \int e^x \, dx$
Step 2: Factoring Constants and Applying Standard Integration Formulae
We extract the constant multiplier $2$ from the first integral and apply the Power Rule and the Exponential Rule for integration.
- The Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ [By the Power Rule of Integration].
- The Exponential Rule: $\int e^x \, dx = e^x + C$ [By the definition of the Exponential Integral].
Step 3: Execution of Integration
Applying the rules identified in Step 2:
$\int 2x^2 \, dx = 2 \int x^2 \, dx = 2 \left( \frac{x^{2+1}}{2+1} \right) = 2 \left( \frac{x^3}{3} \right) = \frac{2}{3}x^3$
$\int e^x \, dx = e^x$
Step 4: Synthesis and Addition of the Constant of Integration
Combining the results, we introduce the arbitrary constant of integration $C$, which is necessary for indefinite integrals representing the family of antiderivatives:
$\int (2x^2 + e^x) \, dx = \frac{2}{3}x^3 + e^x + C$
Visual Representation of the Primitive Function
Final Solution: \frac{2}{3}x^3 + e^x + C
More Questions from Class 12 Mathematics Integrals EXERCISE 7.1
- Q1: Find an anti derivative (or integral) of the following functions by the method of inspection. 1. $\sin 2x$
- Q2: Find an anti derivative (or integral) of the following functions by the method of inspection. 2. $\cos 3x$
- Q3: Find an anti derivative (or integral) of the following functions by the method of inspection. 3. $e^{2x}$
- Q4: Find an anti derivative (or integral) of the following functions by the method of inspection. 4. $(ax + b)^2$
- Q5: Find an anti derivative (or integral) of the following functions by the method of inspection. 5. $\sin 2x - 4e^{3x}$
- Q6: Find the following integrals in Exercises 6 to 20: 6. $\int (4e^{3x} + 1) dx$
- Q7: Find the following integrals in Exercises 6 to 20: 7. $\int x^2(1 - \frac{1}{x^2}) dx$
- Q8: Find the following integrals 8. $\int (ax^2 + bx + c) dx$
- Q10: Find the following integrals 10. $\int (\sqrt{x} - \frac{1}{\sqrt{x}})^2 dx$
- Q11: Find the following integrals 11. $\int \frac{x^3 + 5x^2 - 4}{x^2} dx$
- Q12: Find the following integrals 12. $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$
- Q13: Find the following integrals 13. $\int \frac{x^3 - x^2 + x - 1}{x - 1} dx$
- Q14: Find the following integrals 14. $\int (1 - x)\sqrt{x} dx$
- Q15: Find the following integrals 15. $\int \sqrt{x} (3x^2 + 2x + 3) dx$
- Q16: Find the following integrals 16. $\int (2x - 3\cos x + e^x) dx$
- Q17: Find the following integrals 17. $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$
- Q18: Find the following integrals 18. $\int \sec x (\sec x + \tan x) dx$
- Q19: Find the following integrals 19. $\int \frac{\sec^2 x}{\csc^2 x} dx$
- Q20: Find the following integrals 20. $\int \frac{2 - 3\sin x}{\cos^2 x} dx$
- Q21: Choose the correct answer 21. The anti derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
- Q22: Choose the correct answer 22. If $\frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}$ such that $f(2) = 0$. Then $f(x)$ is
CBSE Solutions for Class 12 Mathematics Integrals
- 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
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Integrals
Top Tutor
Doddaballapura, Bangalore
8 yrs of
Exp
1 student
12 hrs of classes completed
500
per hour
Online Classes
I am a Teacher. i am giving home tuition since 2014.i am certified in commerce.my key skills are teaching in accountancy, economics $ business mathematics for H S. Mathematics for upper primary for any board. Hindi $Bangali literature for 1 to 10.proficient in using computer and the internet for educating. Skilled in conducting fun, engaging and interactive lessons' have attended two national seminar.i have experience to attending class as a guest teacher in various school.
A very good teacher with lots of knowledge about the subject. Not only focuses on getting good marks or being good in academics but focuses on the capacity of the student to learn beyond the textbooks.
Super Tutor
Bannerghatta Main Road, Bangalore
12 yrs of
Exp
49 students
1306 hrs of classes completed
300
per hour
Online Classes
Tutor's home
I am a dedicated tutor. I am having an experience of 12years in teaching accountancy through online as well as privately. I am teaching accountancy for 11, 12, B.Com, B.B.A, ACS, CA ICWA, CMA and all accountancy related courses. My students have fared well in their exams. My Approach is: 1.I will be teaching according to the students way. 2. I will clarify with the students having understood the topics and then only I will proceed to next topic. 3. Regular assessments given to students to know their progress.
Excellent coaching more patience and well experienced teaching my daughter now ready to learn happy. I think she get better knowledge of accountancy from shrimathi mam.
Top Tutor
New Friends Colony, Delhi
10 yrs of
Exp
17 students
4677 hrs of classes completed
500
per hour
Online Classes
I help students in Classes 11–12 develop strong conceptual understanding, advanced problem-solving skills, and confidence for school, board, and competitive-level examinations. My classes cover: Algebra – equations, sequences & series, permutations & combinations, binomial theorem, complex numbers, etc. Trigonometry – identities, equations and applications Coordinate Geometry – straight lines, circles and conic sections Calculus – limits, continuity, differentiation, applications of derivatives, integration and related concepts Probability & Statistics Vectors & 3D Geometry Matrices & Determinants Application-based and higher-order problems Regular practice, tests, and detailed error analysis Exam-oriented preparation with focus on accuracy, speed, and time management My Teaching Approach Instead of simply teaching formulas, I focus on helping students understand why a concept works, when to apply it, and how to approach unfamiliar questions. Concept → Understand → Practise → Apply → Solve → Analyse → Improve Classes are personalised according to the student's current level, curriculum, learning gaps, and academic goals, with structured practice and continuous feedback. Suitable for: CBSE, ICSE, IB, IGCSE, GCSE, Cambridge, and other international curricula.
He explains all the concepts extremely well, and even gives practice questions if we are struggling with any topic. An extremely calm and composed teacher who answers all questions. I saw a huge improvement in my maths scores because of Ravi sir, and also gained a genuine interest in mathematics.
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.
Top Tutor
Mahipalpur, Delhi
14 yrs of
Exp
2 hrs of classes completed
Student's Home
As an experienced teacher for 12 years, I have been teaching to make students interested in the subjects of Maths. My attitude is to help children improve in studies in the best possible way.
