Integrating Powers of x (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Integrating powers of x

How do I integrate powers of x?

• Powers of$x$ are integrated according to the following formulae:

  • If$f\left(x\right)=x^n$ then$\int f\left(x\right)\mathrm{d}x=\frac{x^{n+1}}{n+1}+c$ where$n\in \mathrm{\mathbb{Q}},n\ne -1$ and$c$ is the constant of integration

  •  For each term …

    • … increase the power (of x) by 1

    • … divide by the new power

  • This does not apply when the original power is -1

    • the new power would be 0 and division by 0 is undefined

• If the power of$x$ is multiplied by a constant then the integral is also multiplied by that constant

  • If$f\left(x\right)=a{x}^n$ then$\int f\left(x\right)\mathrm{d}x=\frac{a{x}^{n+1}}{n+1}+c$ where$n\in \mathrm{\mathbb{Q}},n\ne -1$ and$a$ is a constant and$c$ is the constant of integration

• Remember the special case:

  • $\int a\mathrm{d}x=ax+c$

    • e.g. $\int 4\mathrm{d}x=4x+c$ 

  • This allows constant terms to be integrated

How do I integrate expressions containing powers of x?

• The formulae for integrating powers of$x$ apply to all rational numbers so it is possible to integrate any expression that is a sum or difference of powers of$x$

  • e.g.  If$f\left(x\right)=8x^3-2x+4$ then$\int f\left(x\right) \mathrm{d}x=\frac{8x^{3+1}}{3+1}-\frac{2x^{1+1}}{1+1}+4x+c=2x^4-x^2+4x+c$

• Functions involving roots will need to be rewritten as fractional powers of$x$ first

  • eg. If$f\left(x\right)=5\sqrt[3]{x}$ then rewrite as$f\left(x\right)=5x^{\frac{1}{3}}$ and integrate

• Functions involving fractions with denominators in terms of$\mathit{x}$ will need to be rewritten as negative powers of$x$ first

  • e.g.  If$f\left(x\right)=\frac{4}{x^2}+x^2$ then rewrite as$f\left(x\right)=4x^{-2}+x^2$ and integrate   

• Products and quotients cannot be integrated this way so would need expanding/simplifying first

  • e.g.  If$f\left(x\right)=8x^2(2x-3)$ then$\int f\left(x\right)\mathrm{d}x=\int (16x^3-24x^2) \mathrm{d}x=\frac{16x^4}{4}-\frac{24x^3}{3}+c=4x^4-8x^3+c$

Finding the constant of integration

How do I find the constant of integration? 

• STEP 1 

  • Rewrite the function into a more easily integrable form

    • Each term needs to be a power of x (or a constant)

• STEP 2 

  • Integrate each term and remember “+c”

    • Increase power by 1 and divide by new power

• STEP 3

  • Substitute the coordinates of a given point in to form an equation in c

    • Solve the equation to find c