A-level Mathematics/Edexcel/Core 1/Integration

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Basics of integration[edit]

Integration is the opposite of differentiation. For a power of x, you add 1 to the power, divide by the new power and add c, the constant of integration. Note that this rule will not work when the power of x is -1, this requires more advanced methods. The constant of integration is required because if a constant (i.e. a number without x in it) is differentiated it will become zero, and from just integration there is no way to determine the value of this constant.

For example:

 \int 2x \,\, dx


\displaystyle y = x^2 + c

Integrating fractions[edit]

Fractions with an x term in the denominator cannot be integrated as they are; the x term must be brought up to the working line. This can be done easily with the laws of indices.

For example:

              \int \frac{2}{x^2} \,\, dx =  \int 2x^{-2} \,\, dx

Determining the value of c[edit]

You may be given a point on a curve and asked to determine the value of the constant of integration, c. This is quite simple, as the point is given as (x,y); the values of x and y can be plugged in and the equation solved for c.

Worked example:

The the gradient of the curve c is given by \frac{dy}{dx} = 2x.
The point (3,12) lies on c. Hence, find the equation for c.
y= \int 2x \,\, dx
 y = x^2 + c
Plug in values x = 3, y = 12.
12 = 3^2 + c
12 - 9 = c
3 = c
y = x^2 + 3