A-level Maths · Maths
Integration: a clear study guide
Integration reverses differentiation and can also represent accumulated change. Apply the power rule carefully, include the constant for indefinite integrals and use limits for a definite area.
The reverse power rule
For a term ax^n, integrate by increasing the power by one and dividing by the new power: integral ax^n dx = ax^(n+1)/(n+1) + C, provided n is not -1. Integrate each term separately.
The constant C represents the family of antiderivatives. Include it for an indefinite integral and use a known point to find its value when required.
- Increase the power by one.
- Divide by the new power.
- Add C for an indefinite integral.
Definite integrals and area
For a definite integral, substitute the upper and lower limits into an antiderivative and subtract lower from upper. The result is a signed area; if the graph is below the axis, use the context to decide whether a positive geometric area is required.
Sketch the curve or inspect its sign before interpreting the answer. Split the interval if the curve crosses the axis and the question asks for total area.
Connect derivative and integral
A derivative can describe a rate, while an integral accumulates that rate over an interval. State what the value means in the problem and keep units consistent, especially when finding displacement from velocity.
Quick check
What is the integral of 6x^2 with respect to x?
Answer: 2x^3 + C.
What order is used for a definite integral subtraction?
Answer: Evaluate the antiderivative at the upper limit and subtract its value at the lower limit.
Common questions
Why do I add C when integrating?
Differentiating any constant gives zero, so an indefinite integral represents a family of functions that differ by a constant.
How do I find total area when a graph crosses the x-axis?
Split the interval at the crossing, find each geometric area as a positive value and add them.