Introduction to Integration (DP IB Analysis & Approaches (AA): SL): Revision Note

Introduction to integration

What is integration?

• Integration is the inverse (or 'opposite') to differentiation

  • Integration is referred to as antidifferentiation

  • The result of integration is referred to as the antiderivative

• Integration is the process of finding the expression of a function (antiderivative) from an expression of its derivative (gradient function)

What is the notation for integration?

• An integral is normally written in the form $\int f\left(x\right)\mathrm{d}x$

  • the large operator $\int$ means “integrate”

  • “$\mathrm{d}x$” indicates which variable to integrate with respect to

    • In this case it is integrate with respect to $x$

  • $f\left(x\right)$ is the function to be integrated (sometimes called the integrand)

• The antiderivative is sometimes denoted by $F\left(x\right)$

  • Then there’s no need to keep writing the whole integral; refer to it as $F\left(x\right)$

• $F\left(x\right)=\int f\left(x\right)\mathrm{d}x$ may also be called the indefinite integral of$f\left(x\right)$

• $\frac{\mathrm{d}y}{\mathrm{d}x}$ notation can also be used

  • So instead of integrating $f\left(x\right)$ to find its antiderivative $F\left(x\right)$

  • you can think of integrating $\frac{\mathrm{d}y}{\mathrm{d}x}$ to find an expression for its antiderivative $y$

What is the constant of integration? 

• Recall one of the special cases from Differentiating Powers of x

  • If$f\left(x\right)=a$ then$f^{\text{'}}\left(x\right)=0$

• This means that integrating 0 will produce a constant term in the antiderivative

  • Every function, when integrated, potentially has a constant term

• This is called the constant of integration and is usually denoted by the letter $c$

  • it is often referred to as “plus $c$”

• Without more information it is impossible to deduce the value of this constant

  • There are endless antiderivatives, $F\left(x\right)$, for a function$f\left(x\right)$

  • Each one corresponds to a different possible value of $c$