If you know that \( g\ '(x) = 2x \), a natural question to ask is: What is \( g(x) \)? A little experimentation shows that \( g(x) = x^2 \) will work.
In general, if \( f(x) \) is derived from \( F(x) \), i.e. \( f(x)=F\ '(x) \) then \( F(x) \) is called an anti-derivative of f(x). It is also called an indefinite integral of f(x).
The process of finding an anti-derivative of f(x) is denoted by \[ \int f(x)dx \] which means we want to find the \( F(x) \) that \( f(x) \) is derived from. f(x) is referred to as the integrand, i.e. the expression to be integrated.
Example To find \[ \int x^2dx \] you apply the reverse of the steps you use for differentiation, so first add 1 to the given exponent and and divide by the new exponent to get \[ \int x^2dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}\] However, \[ \frac{d}{dx}(\frac{x^3}{3} + c) = x^2 \] so there needs to be an extra step of adding a constant c to get the full solution \[ \int x^2dx = \frac{x^3}{3} + c\] so \[ F(x) = \frac{x^3}{3} + c\]
The constant must always be included when finding an anti-derivative because anti-derivatives are defined only up to the constant.
Repeat the above process for \(\int x^3 dx\).
This process applies to more functions than simple powers. For example, if you have been told that \[ \frac{d}{dx}\sin x = \cos x \] then setting \( F\ '(x) = \cos x = f(x) \) you know immediately that \[ \int \cos x dx = \sin x + c \] i.e. to find an indefinite integral of f(x), just reverse the differentiation that produced f(x) then add a constant.
Use your knowledge of derivatives to find \(\int \sec^2 x dx\).
Guided Examples
O E Qs
Since finding an indefinte integral is 'undoing' a differentiation, you would expect anti-derivatives to have some of the properties of derivatives. The two most important are:
If \( h(x) = f(x) + g(x) \) then \[ \int h(x) = \int f(x) + \int g(x)\quad \ \]
If \( h(x) = cf(x) \) where c is a real constant, then \[ \int h(x) = c \int f(x) \quad \ \]
The set of steps for integrating a power is referred to as an integration rule. Other integration rules can be derived from the rules for differentiation. The next lesson provides more detail.
The rule for integrating a power is:
If \( f(x) = x^n \) and \( n \ne -1 \) then \[ \int f(x) = \frac{x^{n+1}}{n+1} + c \] where n is an integer or rational number.
Example To illustrate how the rules work together, to evaluate \[ \int ((4x)^{1/2} + 4x^3) dx \] Set \[ \begin{align} I &= \int ((4x)^{1/2} + 4x^3) dx \quad \\ &= \frac{4x^{1/2+1}}{1/2+1} + 4x^4/4 + c \quad \\ &= \frac{8}{3}x^{3/2} + x^4 + c \quad \\ \end{align} \ \]
Find \(\int 2(x^3+\sec^2 x) dx\).
The basic strategy for finding indefinite integrals is to apply the properties and integration rules to the integrand to produce an indefinite integral.
In many cases, the integrand will not be in a form you can immediately apply the rules to. Then you need to change the form of the integrand, i.e. rewrite the integrand, so you can apply the rules to it.