Antiderivatives and Integration Rules
An antiderivative reverses differentiation.
About this topic
An antiderivative of f is a function F with F′ = f; the full family is F(x) + C, where C is the constant of integration. Rules: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1), ∫eˣ dx = eˣ + C, ∫1/x dx = ln of the absolute value of x + C, ∫cos x dx = sin x + C.
Integrals are linear, so constants pull out and sums can be integrated term by term. For ∫(6x² − 4x + 5) dx the answer is 2x³ − 2x² + 5x + C. You can always check an answer by differentiating it, which should return the original function. A condition like F(0) = 3 fixes the value of C.
The substitution method reverses the chain rule: in ∫2x cos(x²) dx, let u = x², so du = 2x dx and the integral becomes ∫cos u du = sin(x²) + C. Other trig results include ∫sin x dx = −cos x + C.
Antiderivatives recover position from velocity or total cost from marginal cost. Common mistakes are forgetting the + C, mishandling the exponent rule for n = −1, and failing to adjust the constants in a substitution. Practice recognising which pattern fits.
Key ideas
- ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
- Always add + C
- ∫ eˣ dx = eˣ + C
Quick check
Use cases: where Antiderivatives and Integration Rules helps
- Homework: review the key idea behind Antiderivatives and Integration Rules before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Antiderivatives and Integration Rules without notes.
- Visual learning: play with the interactive lab to see how Antiderivatives and Integration Rules changes when you adjust its inputs.
- Teaching a friend: explain Antiderivatives and Integration Rules in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Antiderivatives and Integration Rules?
An antiderivative reverses differentiation.
What will I learn in Antiderivatives and Integration Rules?
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. Always add + C. ∫ eˣ dx = eˣ + C.
Is Antiderivatives and Integration Rules part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Antiderivatives and Integration Rules lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Antiderivatives and Integration Rules?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
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