Unit 4: Integral calculus
Mathematics-II notes · PTU syllabus (BSIT202/BSBC202)
On this page
Unit summary
Integration accumulates quantities — areas, totals and costs. This unit covers indefinite integrals, integration by substitution, by parts and by partial fractions, definite integrals, and numerical integration by the trapezoidal rule and Simpson's 1/3 and 3/8 rules.
After this unit you can
- Find indefinite integrals using standard results
- Integrate by substitution, parts and partial fractions
- Evaluate definite integrals
- Apply numerical integration rules
PTU syllabus topics
- Indefinite integral
- integration by substitution/parts/partial fractions
- definite integral
- numerical integration — trapezoidal rule
- Simpson's 1/3 and 3/8 rules
Power rule
∫xⁿ dx = xⁿ⁺¹ / (n + 1) + C
By parts
∫u dv = uv − ∫v du
Trapezoidal rule
(h/2) [y0 + 2(y1 + … + yn−1) + yn]
Simpson's 1/3 rule
(h/3) [y0 + 4(odd) + 2(even) + yn]
Topic 1
Indefinite integrals
Power
∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, n ≠ −1
Reciprocal
∫ (1/x) dx = log x + C
Exponential
∫ eˣ dx = eˣ + C
Trigonometric
∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C
Topic 2
Integration by substitution
- Replace part of the integrand by u so that du appears.
Example
∫ 2x (x² + 1)⁵ dx: let u = x² + 1, du = 2x dx; ∫ u⁵ du = u⁶/6 + C = (x² + 1)⁶/6 + C.
Topic 3
Integration by parts
Rule
∫ u dv = uv − ∫ v du
Choosing u (ILATE)
Inverse trig, logarithmic, algebraic, trigonometric, exponential
Example
∫ x eˣ dx: u = x, dv = eˣ dx → x eˣ − ∫ eˣ dx = eˣ (x − 1) + C.
Topic 4
Integration by partial fractions
- Split a rational function into simpler fractions, then integrate each.
Example
∫ dx ÷ [(x − 1)(x + 2)]: 1 ÷ [(x − 1)(x + 2)] = (1/3) ÷ (x − 1) − (1/3) ÷ (x + 2); integral = (1/3) log[(x − 1) ÷ (x + 2)] + C.
Topic 5
Definite integrals
Fundamental theorem
∫ from a to b f(x) dx = F(b) − F(a)
Property
∫ from a to b = −∫ from b to a
Example
∫ from 0 to 2 of 3x² dx = [x³] from 0 to 2 = 8.
Topic 6
Numerical integration
Trapezoidal rule
(h/2) [y0 + 2(y1 + y2 + … + yn−1) + yn]
Simpson's 1/3 rule (n even)
(h/3) [y0 + 4(odd-indexed ordinates) + 2(even-indexed interior ordinates) + yn]
Simpson's 3/8 rule (n multiple of 3)
(3h/8) [y0 + 3(y1 + y2 + y4 + y5 + …) + 2(y3 + y6 + …) + yn]
Example
∫ from 0 to 4 of x² dx with h = 1 (y = 0, 1, 4, 9, 16): trapezoidal = 0.5 [0 + 2(14) + 16] = 22; Simpson's 1/3 = (1/3) [0 + 4(1 + 9) + 2(4) + 16] = 21.33 — exact value 21.33.
Key terms
- Indefinite integral
- Antiderivative plus a constant
- Definite integral
- Integral between limits giving a number
- Integration by parts
- Technique based on the product rule
- Partial fractions
- Splitting a rational function into simpler terms
- Simpson's rule
- Numerical integration using parabolic arcs
Quick revision
- Standard integrals.
- Substitution; by parts (ILATE); partial fractions.
- Definite integrals and properties.
- Trapezoidal, Simpson's 1/3 and 3/8 rules and their conditions.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Integrate x⁴ with respect to x.
- Q2.State the formula for integration by parts.
- Q3.What is ILATE?
- Q4.Evaluate ∫ from 1 to 2 of 2x dx.
- Q5.When can Simpson's 1/3 rule be applied?
- Q6.State the trapezoidal rule.
Long-answer questions
- Q1.Evaluate integrals by substitution and by parts (numerical).
- Q2.Evaluate an integral using partial fractions (numerical).
- Q3.Evaluate a definite integral and explain its meaning (numerical).
- Q4.Evaluate an integral by the trapezoidal and Simpson's rules and compare (numerical).
Stuck on this unit?
Message SBS on WhatsApp for help with Mathematics-II, or to ask about studying B.Sc IT at Synetic.