Reversing Differentiation | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Find a function from its derivative

Differentiation takes a function and finds its gradient rule. Integration reverses that calculation. If the gradient is $3x^2$ , a possible original function is $x^3$ , because differentiating it gives $3x^2$ .

But $x^3+7$ gives the same derivative. Any added constant disappears when differentiated, so the general answer must include an unknown constant $c$ .

\int 3x^2\,dx=x^3+c

The symbol $\int$ means integrate. The expression being integrated is the integrand; $dx$ says that the variable is $x$ . Without integration limits, this is an indefinite integral: a family of possible functions, not one number.

Increase the power, then divide

To obtain $x^2$ after differentiating, start with a cubic. Its derivative introduces a factor of three, so divide by three first:

\frac{d}{dx}\left(\frac{x^3}{3}\right)=\frac13\times3x^2=x^2

The same reasoning works for a rational power $n$ (an integer or a fraction), except $n=-1$ :

\begin{gathered}\int x^n\,dx=\frac{x^{n+1}}{n+1}+c\\n\ne-1\end{gathered}

At $n=-1$ , the proposed divisor is zero, so this rule cannot be used. The power rule is supplied in MF19; the sheet omits arbitrary constants, but your indefinite answer still needs $+c$ .

Worked example

Integrate 6x² − 4x + 5.

Keep each constant multiplier and integrate the terms separately. A constant such as 5 integrates to $5x$ , since the derivative of $5x$ is 5.

\begin{gathered}\int(6x^2-4x+5)\,dx\\=\frac63x^3-\frac42x^2+5x+c\\=2x^3-2x^2+5x+c\end{gathered}

Differentiate the whole answer to check every term. One constant $c$ is enough: several added constants can be combined into one.

Your turnPractice

Find $\int(8x^3-6)\,dx$ . Explain why $24x^2$ is not the answer.

Show worked answer

Answer

2x^4-6x+c

$24x^2$ is the derivative of the integrand. Differentiating our answer instead returns $8x^3-6$ .