1. Indefinite Integrals
These integrals find a family of functions. They always include a constant C.
∫ x² dx = x³/3 + C
What Is an Integral?
An integral is a mathematical tool used to find accumulated quantities, areas under curves, total change, and antiderivatives. If derivatives describe rates of change, integrals help us add up those changes.
Basic idea:
∫ f(x) dx = F(x) + C
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Quick Guide to Integrals
2. Definite Integrals
These calculate the signed area under a curve between two values.
∫ₐᵇ f(x) dx
3. Power Rule
For powers of x, increase the exponent by 1 and divide by the new exponent.
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C
4. Constant Rule
A constant can be pulled outside an integral.
∫ 7x dx = 7∫ x dx
Facts About Integrals
01
Integrals are used in physics to calculate distance, work, energy, pressure, and electric charge.
02
The integral symbol ∫ comes from an elongated letter “S”, standing for “sum”.
03
Definite integrals can represent area, but if the graph goes below the x-axis, the area is counted as negative.
04
Integration and differentiation are inverse processes according to the Fundamental Theorem of Calculus.