Integrating Basic Functions (Edexcel IGCSE Further Pure Maths): Revision Note

Integrating Powers of x

How do I integrate powers of x?

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

  • $\int x^n\mathrm{d}x=\frac{x^{n+1}}{n+1}+c$  (where$c$ is the constant of integration)

    • This is valid for any value of $n$ except $n=-1$

    • So you cannot integrate  $\int x^{-1}\mathrm{d}x=\int \frac{1}{x}\mathrm{d}x$ this way

• If $x^n$ is multiplied by a constant $a$ then

  • $\int a{x}^n\mathrm{d}x=a\int x^n\mathrm{d}x=\frac{a{x}^{n+1}}{n+1}+c$

    • This also is not valid for $n=-1$

• These formulae are not on the exam formula sheet, so you need to remember them

• 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

  • e.g. $\int \sqrt[3]{x}\mathrm{d}x$

    • rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$

    • then integrate

• Fractions with $x$ in the denominator will need to be rewritten as negative powers

  • e.g. $\int \frac{1}{x^2}\mathrm{d}x$

    • rewrite $\frac{1}{x^2}$ as $x^{-2}$

    • then integrate

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

• The formulae can be used to integrate sums or differences of powers of$x$

  • Just integrate term by term

    • e.g. $\int \left(8x^3-2x+4\right) \mathrm{d}x$

    • $=\frac{8x^{3+1}}{3+1}-\frac{2x^{1+1}}{1+1}+4x+c$

    • $=2x^4-x^2+4x+c$
               

• Products and quotients cannot be integrated this way

  • You need to expand and/or simplify first

    • e.g. $\int 8x^2\left(2x-3\right)\mathrm{d}x$

    • expand $8x^2\left(2x-3\right)$ as $16x^3-24x^2$

    • then integrate term by term

    • you cannot just multiply the integrals of $8x^2$ and $2x-3$ together

What might I be asked to do once I’ve integrated?

• You may be given the derivative of a function and asked to find the function

  • Integration and differentiation are inverse operations so

    • $\int {\mathrm{f}}^{\text{'}}\left(x\right)\mathrm{d}x=\mathrm{f}\left(x\right)+c$

    • $\int \frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x=y+c$

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

• The area under a curve can also be found using integration

Integrating Trig Functions

How do I integrate sin and cos?

• You can integrate $\sin x$ and $\cos x$ by using the formulae

  • $\mathbf{\int }\mathbf{sin}\mathit{x}\mathbf{d}\mathit{x}\mathbf{=}\mathbf{-}\mathbf{cos}\mathit{x}\mathbf{+}\mathit{c}$

  • $\mathbf{\int }\mathbf{cos}\mathit{x}\mathbf{d}\mathit{x}\mathbf{=}\mathbf{sin}\mathit{x}\mathbf{+}\mathit{c}$

    • $c$ is the constant of integration

• If $x$ is multiplied by a constant $a$ then

  • $\mathbf{\int }\mathbf{sin}\mathit{a}\mathit{x}\mathbf{d}\mathit{x}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathit{a}}\mathbf{cos}\mathit{a}\mathit{x}\mathbf{+}\mathit{c}$

  • $\mathbf{\int }\mathbf{cos}\mathit{a}\mathit{x}\mathbf{d}\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathit{a}}\mathbf{sin}\mathit{a}\mathit{x}\mathbf{+}\mathit{c}$

• None of these formulae are on the exam formula sheet, so you need to remember them

• For calculus with trigonometric functions angles must be measured in radians

  • Make sure you know how to change the angle mode on your calculator

Integrating e^x

How do I integrate exponentials?

• ${\mathrm{e}}^x$ can be integrated using the formula

  • $\mathbf{\int }{\mathbf{e}}^{\mathit{x}}\mathbf{d}\mathit{x}\mathbf{=}{\mathbf{e}}^{\mathit{x}}\mathbf{+}\mathit{c}$

    • $c$ is the constant of integration

• If $x$ is multiplied by a constant $a$ then

  •  $\mathbf{\int }{\mathbf{e}}^{\mathit{a}\mathit{x}}\mathbf{d}\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathit{a}}{\mathbf{e}}^{\mathit{a}\mathit{x}}\mathbf{+}\mathit{c}$

• These formulae are not on the exam formula sheet, so you need to remember them