Integrating Powers of x (DP IB Applications & Interpretation (AI): HL): Revision Note

Integrating powers of x

How do I integrate powers of x?

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

  • 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

• This is given in the formula booklet

• If the power of$x$ term 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$n\in \mathrm{\mathbb{Q}},n\ne -1$

    • $a$ 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

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

  • e.g. 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    

How do I integrate sums and differences of powers of x?

• To integrate a sum or difference of power of$x$ terms, just integrate term by term

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

• Products and quotients cannot be integrated this way so would need to be expanded/simplified first

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

What might I be asked to do once I’ve found the anti-derivative (integrated)?

• With more information the constant of integration,$c$, can be found

• The area under a curve can be found using integration