Definite integral evaluation and properties
Use a primitive and ordered limits; preserve signs when transforming an integral
\[\int_a^b f(x)\,dx=F(b)-F(a)\]
- Reverse or repeat limits Swapping the limits negates the value. Equal limits give zero
- Split an interval Add the signed contributions over adjacent intervals
- Use linearity Apply each constant to its own integral and add the results
- Rename consistently A dummy-variable rename changes neither the integrand rule nor the limits
- Check an unknown endpoint Solve the primitive-difference equation and retain only roots in the stated domain. Use monotonicity when claiming uniqueness
Worked example
Evaluate the displayed integral, then reverse its limits
\[\int_1^4(2x+3)\,dx\]
- \(\displaystyle F(x)=x^2+3x\)Reverse the power rule term by term
- \(\displaystyle F(4)=28,\quad F(1)=4\)Evaluate the chosen primitive at both ordered endpoints
- \(\displaystyle \int_1^4(2x+3)\,dx=28-4=24\)Upper value minus lower value; a constant would cancel
- \(\displaystyle \int_4^1(2x+3)\,dx=-24\)Reversing the limits changes the sign
Common mistakes
Subtracting in the wrong order
✕\(\displaystyle \int_a^b f(x)\,dx=F(a)-F(b)\)✓\(\displaystyle \int_a^b f(x)\,dx=F(b)-F(a)\)
Use the upper-endpoint primitive value firstKeeping an arbitrary constant
✕\(\displaystyle \int_1^4(2x+3)\,dx=24+C\)✓\(\displaystyle \int_1^4(2x+3)\,dx=24\)
The same C occurs at both endpoints and cancelsIgnoring a negative scale
✕\(\displaystyle 2(24)-3(-5)=33\)✓\(\displaystyle 2(24)-3(-5)=63\)
Subtracting a negative contribution adds its magnitudeDifficulty levels
introForm a primitive and evaluate with nonnegative integer endpoints, or recognise zero width
standardReuse supplied-primitive evaluation with signed/rational endpoints and practise dummy-variable renaming
challengeApply reversal, splitting and linearity, or solve a unique endpoint problem
examSelect a correct property or substitution line, then give the exact value or endpoint
Questions about this topic
Does a correct choice prove full independent integration skill?+
No. Choices show recognition and coefficient blanks show local values. Dependent steps do not establish independent written-method mastery
Is this part of the required Units 1–2 course?+
This placement is Methods Units 3–4. The optional Units 1–2 definite-integration extension is not activated by this packet
Are transcendental and rational-power integrands added here?+
No. This generator uses polynomials. Existing transcendental integration resources retain their own questions and exact-response contracts
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 8: Integration
In this chapter
Approximate area by an endpoint or trapezium ruleEvaluate and transform a definite integralElementary primitives and exact definite integralsFind complete families of polynomial anti-derivativesThe area function and the fundamental theorem of calculusSigned integrals, geometric area and motionFind the area between two curvesAverage value of a functionTotal change and kinematics from a rateDetermine a polynomial from its derivative and a boundary conditionRecover motion from acceleration