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• A-Level Maths

# Integration

Before going too far down this notes sheet, ensure you're on top form for everything differentiation... else this will be harder than it needs to be!

Integration is the opposite of differentiation, and can be used to find the initial function form its derivative. For example, to find f(x) from f'(x), you integrate. It is also commonly used to find the area between a curve and the x-axis.

## Integrating xⁿ

When differentiating xⁿ, you multiply by the power and then reduce the power by one. The reverse of this is to add one to the power and divide by this:

If f'(x) = xⁿ, then f(x) = (xⁿ⁺¹) / (n+1)

If there is a coefficient of x, you do the same but:

If f'(x) = kxⁿ, then f(x) = k(xⁿ⁺¹) / (n+1)

Note the difference between differentiating and integrating xⁿ:

• Just like when differentiating polynomials (functions with multiple terms), integrate one term at a time.

## Definite & Indefinite Integrals

Integration is often noted using ∫ () dx, where the elongated S tells you to integrate the function in the brackets while the dx tells you to do so with respect to x. It is important to get this notation right.

• When there are no limits with the ∫, the integral is called indefinite, because it can only give a function with an unknown y-intercept, c.

• If there are limits attached to the ∫, a and b, the integral is definite, as it produces a value for a defined interval [a, b].