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:

${\displaystyle \int 2x\,\,dx}$

becomes:

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

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:

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

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 ${\displaystyle (x,y)}$; 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 ${\displaystyle {\frac {dy}{dx}}=2x}$.

The point ${\displaystyle (3,12)}$ lies on c. Hence, find the equation for c.

${\displaystyle y=\int 2x\,\,dx}$

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

Plug in values x = 3, y = 12.

${\displaystyle 12=3^{2}+c}$

${\displaystyle 12-9=c}$

${\displaystyle 3=c}$

${\displaystyle y=x^{2}+3}$