Find the best tutors and institutes for Class 12 Tuition
Q3:
Evaluate the definite integrals in Exercises 1 to 20.
3. $\int_{1}^{2} (4x^3 - 5x^2 + 6x + 9) dx$
Solution :
Step 1: Application of the Fundamental Theorem of Calculus
To evaluate the definite integral of the polynomial function $f(x) = 4x^3 - 5x^2 + 6x + 9$ over the interval $[1, 2]$, we utilize the Second Fundamental Theorem of Calculus, which states that if $F(x)$ is the antiderivative of $f(x)$, then:
$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ [Per the Fundamental Theorem of Calculus, Part II]
Step 2: Determining the Antiderivative
We apply the power rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, to each term of the integrand individually:
- $\int 4x^3 \, dx = 4 \left( \frac{x^4}{4} \right) = x^4$
- $\int -5x^2 \, dx = -5 \left( \frac{x^3}{3} \right) = -\frac{5}{3}x^3$
- $\int 6x \, dx = 6 \left( \frac{x^2}{2} \right) = 3x^2$
- $\int 9 \, dx = 9x$
Thus, the antiderivative is $F(x) = x^4 - \frac{5}{3}x^3 + 3x^2 + 9x$.
Step 3: Evaluating at Boundaries
We evaluate $F(x)$ at the upper bound ($x=2$) and lower bound ($x=1$):
For $x = 2$:
$F(2) = (2)^4 - \frac{5}{3}(2)^3 + 3(2)^2 + 9(2)$
$F(2) = 16 - \frac{5}{3}(8) + 3(4) + 18$
$F(2) = 16 - \frac{40}{3} + 12 + 18 = 46 - \frac{40}{3}$
$F(2) = \frac{138 - 40}{3} = \frac{98}{3}$
For $x = 1$:
$F(1) = (1)^4 - \frac{5}{3}(1)^3 + 3(1)^2 + 9(1)$
$F(1) = 1 - \frac{5}{3} + 3 + 9 = 13 - \frac{5}{3}$
$F(1) = \frac{39 - 5}{3} = \frac{34}{3}$
Step 4: Calculating the Definite Integral
Subtracting $F(1)$ from $F(2)$:
$\int_{1}^{2} f(x) \, dx = \frac{98}{3} - \frac{34}{3}$
$\int_{1}^{2} f(x) \, dx = \frac{64}{3}$
Visualization of the Definite Integral
Final Solution: \frac{64}{3}
More Questions from Class 12 Mathematics Integrals EXERCISE 7.8
- Q1: Evaluate the definite integrals in Exercises 1 to 20. 1. $\int_{-1}^{1} (x+1) dx$
- Q2: Evaluate the definite integrals in Exercises 1 to 20. 2. $\int_{2}^{3} \frac{1}{x} dx$
- Q4: Evaluate the definite integrals in Exercises 1 to 20. 4. $\int_{0}^{\frac{\pi}{4}} \sin 2x dx$
- Q5: Evaluate the definite integrals in Exercises 1 to 20. 5. $\int_{0}^{\frac{\pi}{2}} \cos 2x dx$
- Q6: Evaluate the definite integrals in Exercises 1 to 20. 6. $\int_{4}^{5} e^x dx$
- Q7: Evaluate the definite integrals in Exercises 1 to 20. 7. $\int_{0}^{\frac{\pi}{4}} \tan x dx$
- Q8: Evaluate the definite integrals in Exercises 1 to 20. 8. $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc x dx$
- Q9: Evaluate the definite integrals in Exercises 1 to 20. 9. $\int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}$
- Q10: Evaluate the definite integrals in Exercises 1 to 20. 10. $\int_{0}^{1} \frac{dx}{1+x^2}$
- Q11: Evaluate the definite integrals in Exercises 1 to 20. 11. $\int_{2}^{3} \frac{dx}{x^2-1}$
- Q12: Evaluate the definite integrals in Exercises 1 to 20. 12. $\int_{0}^{\frac{\pi}{2}} \cos^2 x dx$
- Q13: Evaluate the definite integrals in Exercises 1 to 20. 13. $\int_{2}^{3} \frac{x dx}{x^2+1}$
- Q14: Evaluate the definite integrals in Exercises 1 to 20. 14. $\int_{0}^{1} \frac{2x+3}{5x^2+1} dx$
- Q15: Evaluate the definite integrals in Exercises 1 to 20. 15. $\int_{0}^{1} x e^{x^2} dx$
- Q16: Evaluate the definite integrals in Exercises 1 to 20. 16. $\int_{1}^{2} \frac{5x^2}{x^2+4x+3} dx$
- Q17: Evaluate the definite integrals in Exercises 1 to 20. 17. $\int_{0}^{\frac{\pi}{4}} (2\sec^2 x + x^3 + 2) dx$
- Q18: Evaluate the definite integrals in Exercises 1 to 20. 18. $\int_{0}^{\pi} (\sin^2 \frac{x}{2} - \cos^2 \frac{x}{2}) dx$
- Q19: Evaluate the definite integrals in Exercises 1 to 20. 19. $\int_{0}^{2} \frac{6x+3}{x^2+4} dx$
- Q20: Evaluate the definite integrals in Exercises 1 to 20. 20. $\int_{0}^{1} (x e^x + \sin \frac{\pi x}{4}) dx$
- Q21: Choose the correct answer in Exercises 21 and 22. 21. $\int_{1}^{\sqrt{3}} \frac{dx}{1+x^2}$ equals
- Q22: Choose the correct answer in Exercises 21 and 22. 22. $\int_{0}^{\frac{2}{3}} \frac{dx}{4+9x^2}$ 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 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.7
- 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
T Nagar, Chennai
10 yrs of
Exp
6 students
1332 hrs of classes completed
700
per hour
Online Classes
I have over 10 years of experience teaching CBSE Class 12 Biology and Chemistry, helping students build a strong conceptual foundation while achieving excellent board examination results. My teaching methodology focuses on simplifying complex topics through clear explanations, logical reasoning, and regular practice. Every chapter is taught systematically from the NCERT textbook, ensuring students understand every concept thoroughly before progressing to application-based and board-level questions. My classes include detailed concept explanations, NCERT line-by-line learning, important diagrams, numerical problem-solving (Chemistry), previous years' board questions, competency-based questions, and chapter-wise assessments. I provide personalized attention by identifying each student's learning needs and adapting my teaching pace accordingly. Regular doubt-solving sessions, revision classes, mock tests, and performance analysis help students improve accuracy, confidence, and time management before the board examinations. My objective is not only to help students score high marks in the CBSE Class 12 Board Examination but also to develop a deep understanding of Biology and Chemistry, enabling them to confidently prepare for competitive examinations such as NEET. My classes are interactive, well-structured, and focused on helping every student achieve their academic goals through consistent guidance and disciplined practice.
Excellent coaching. She uses simple and easy methods to help students understand complex concepts. Thank you, ma'am, for your support in building my daughter's future.
Top Tutor
Mulund West, Mumbai
10 yrs of
Exp
1 student
8 hrs of classes completed
700
per hour
Online Classes
Student's Home
Experienced and dedicated Science educator with a strong background in teaching Physics, ,Maths to Grade 12 students. Skilled in creating engaging lesson plans aligned with CBSE/ICSE/State Board curricula, fostering scientific curiosity, and using modern teaching aids such as smart boards, simulations, and lab experiments. Proficient in classroom management, differentiated instruction, and continuous student assessment. Committed to promoting STEM education and preparing students for board exams and competitive entrance tests.
Explanation for every chapter was on point and the notes given helped a lot and made easier to understand the concepts.
Top Tutor
Edapally, Kochi
4 yrs of
Exp
48 hrs of classes completed
500
per hour
Student's Home
Tutor's home
Qualified french teacher and tutor with a master degree in french from delhi university. Working as a language expert in ernst and young. Qualified. Have 4 years of teaching experience. With c1 certification.
Super Tutor
Arrah Chowk, Arrah
11 yrs of
Exp
23 students
826 hrs of classes completed
400
per hour
Online Classes
A Highly talented Chemistry teacher with excellent communication skills demonstrated by 11 years of teaching experience. Strong theoretical and good knowledge of Chemistry. A good Communication skills to maintain great relationship with students and ability to work well with others. You Can also watch me on my YouTube channel L.k.Soni Chemistry Classes. During live classes developed an approach where students can focus on the classes without any Distraction.
I had the opportunity to attend one-on-one Classes for class 12 CBSE Chemistry with him, and the experience was excellent. His way of explaining concepts is very clear, simple, and easy to understand. He makes sure that every doubt is addressed patiently and focuses not just on memorising reactions and formulas, but on developing a proper understanding of the concepts. I especially appreciated his exam-oriented approach, which makes difficult topics feel much more manageable and helps build confidence for the CBSE Board exams. He is very knowledgeable, supportive, and genuinely dedicated to his students. I would highly recommend his classes to any Class 12 CBSE Chemistry student looking for strong conceptual clarity and effective preparation.
Top Tutor
Kushaiguda, Hyderabad
13 yrs of
Exp
1 student
126 hrs of classes completed
450
per hour
Online Classes
Student's Home
Tutor's home
Teaching is my passion I am into teaching profession since 16 years. I like challenges with young minds.. I will make students to confident by explaining concepts, taking special care. Cent percent assurance to child growth, committed to work towards target. I used to counsel the students to succeed in their exams.
Good Teaching, With excellent Communication skills & I'm enjoying, I like to attending classes and also I'm understanding.
