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## Integration, the inverse of differentiation

Integration can be used to reverse the effect of differentiation.

The rule for differentiation is: if $y=ax^n$ then $\frac{dy}{dx}=anx^{n-1}$

So we multiply by the power and then take one off the power.

If we reverse this process we get add one to the power and divide by this new power.

So if $\frac{dy}{dx}=ax^n$ then $y=\frac{ax^{n+1}}{n+1}$

Lets consider what happens when we differentiate a constant i.e. $y=c$

This is the same as $y=cx^0\Rightarrow \frac{dy}{dx}=0\times cx^{-1}=0$

So if we differentiate a constant we get zero which means if we integrate zero we get a constant.

So $y=ax^n$ then $\frac{dy}{dx}=anx^{n-1}+c$

Examples

$\frac{dy}{dx}=8x^3+3x^2-5$, what is a general solution for y?

To get y we need to integrate

So $y=\frac{8x^4}{4}+\frac{3x^3}{3}-5x+c$

$\Rightarrow y=2x^4+x^3+5x+c$

This is a general solution because $c$ could have any value.

Here I have given $c$ values that range for -1 to 3, but there were an infinite other possibilities.  The important thing to note is that if we differentiate any of these curves they all give the same expression.

The gradient function of a curve is $\frac{dy}{dx}=6x^2-3$, the curve passes through the point $(1,3)$.

What is the particular solution.

Integrating gives $y=2x^3-3x+c$

Since the curve goes through $(1,3)$ we can substitute in these values to get $3=2-3+c\Rightarrow c=4$

This gives the particular solution $y=2x^3-3x+4$

A particular solution is one where the value of c has been found so that now we can only draw one particular curve.

### Notation

$\int f(x) dx$ means integrate $f(x)$.  The $dx$ means that the variable we are considering is x.

Example

Find the general solution of $y=\int (3x+4) dx$

$y=\frac{3x^2}{2}+4x+c$