# A-level Mathematics/Edexcel/Core 1/Integration

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## 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. This 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 line l has an equation $y = 2x$
Line l is a tangent to the curve c at the point $(3,12)$; find the equation for curve c.
$\int 2x \,\, dx$
$y = x^2 + c$
Plug in values x = 3, y = 12.
$12 = 3^2 + c$
$12 - 9 = c$
$3 = c$
$y = x^2 + 3$