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.


 * $$12 = 3^2 + c$$


 * $$12 - 9 = c$$


 * $$3 = c$$


 * $$y = x^2 + 3$$