A-level Mathematics/Edexcel/Core 1/Integration

Basics of integration

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:

$\int 2x\,\,dx$

becomes:

$\displaystyle y=x^{2}+c$

Integrating fractions

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

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 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.

$12=3^{2}+c$

$12-9=c$

$3=c$

$y=x^{2}+3$