Definite integrals of polynomials: Class 12 practice
∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative of f. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Power rule for derivatives
- Fractions
The idea behind definite integrals of polynomials
∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative of f. For powers, ∫xⁿ dx = xⁿ⁺¹/(n + 1).
Step-by-step method
- Integrate term by term.
- Evaluate at the upper limit, then at the lower limit.
- Subtract: F(b) − F(a).
Worked example 1 (Basic)
Evaluate ∫ from -2 to -1 of (3x² + 2x + 6) dx.
- Antiderivative F(x) = 1x³ + 1x² + 6x
- F(-1) − F(-2) = 10
- Answer: 10
Worked example 2 (Standard)
Evaluate ∫ from -2 to 0 of (4x² + 5x + 6) dx.
- Antiderivative F(x) = 4/3x³ + 5/2x² + 6x
- F(0) − F(-2) = 38/3
- Answer: 38/3
Worked example 3 (Challenge)
Evaluate ∫ from 1 to 4 of (6x² + 0x − 1) dx.
- Antiderivative F(x) = 2x³ + 0x² − 1x
- F(4) − F(1) = 123
- Answer: 123
Common mistakes and misconceptions
- Adding F(a) instead of subtracting
- Forgetting to divide by the new power
Practice questions
- (Basic) Evaluate ∫ from 0 to 1 of (2x² − 2x + 5) dx.
- (Basic) Evaluate ∫ from 0 to 2 of (2x² − 3x + 5) dx.
- (Standard) Evaluate ∫ from -2 to 0 of (3x² + 5x + 7) dx.
- (Standard) Evaluate ∫ from -1 to 0 of (3x² + 2x + 7) dx.
- (Challenge) Evaluate ∫ from -1 to 0 of (6x² + 1x + 6) dx.
- (Challenge) Evaluate ∫ from -2 to 1 of (5x² + 0x + 6) dx.
Answer key with full working
- Q1: Antiderivative F(x) = 2/3x³ − 1x² + 5x → F(1) − F(0) = 14/3 → Answer: 14/3
- Q2: Antiderivative F(x) = 2/3x³ − 3/2x² + 5x → F(2) − F(0) = 28/3 → Answer: 28/3
- Q3: Antiderivative F(x) = 1x³ + 5/2x² + 7x → F(0) − F(-2) = 12 → Answer: 12
- Q4: Antiderivative F(x) = 1x³ + 1x² + 7x → F(0) − F(-1) = 7 → Answer: 7
- Q5: Antiderivative F(x) = 2x³ + 1/2x² + 6x → F(0) − F(-1) = 15/2 → Answer: 15/2
- Q6: Antiderivative F(x) = 5/3x³ + 0x² + 6x → F(1) − F(-2) = 33 → Answer: 33
Syllabus note
This skill is listed under "Integrals" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is definite integrals of polynomials in?
"Integrals", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.