Concept of Antiderivative and Indefinite Integral

Related calculator: Integral (Antiderivative) Calculator with Steps

In the Calculus I (Differential Calculus) our main purpose was to find derivative of given function.

But often we need to solve inverse task: given a function $$${f{{\left({x}\right)}}}$$$ find function $$${F}{\left({x}\right)}$$$ whose derivative is $$${f{{\left({x}\right)}}}$$$. In other words, we need to find function $$${F}{\left({x}\right)}$$$ such that $$${F}'{\left({x}\right)}={f{{\left({x}\right)}}}$$$.

Indeed, suppose we are given velocity of object $$${v}{\left({t}\right)}$$$ and need to find its position function $$${s}{\left({t}\right)}.$$$ Since $$${s}'{\left({t}\right)}={v}{\left({t}\right)}$$$, this leads us to inverse task. Similarly, we may want to find velocity $$${v}{\left({t}\right)}$$$ of object knowing its acceleration $$${a}{\left({t}\right)}$$$. Since $$${v}'{\left({t}\right)}={a}{\left({t}\right)}$$$ we again need to solve inverse task.

Definition. A function $$${F}$$$ is called antiderivative of $$${f{}}$$$ on interval $$${I}$$$ if $$${F}'{\left({x}\right)}={f{{\left({x}\right)}}}$$$ for all $$${x}$$$ in $$${I}$$$.

Example 1. Find antiderivative of $$${f{{\left({x}\right)}}}={{x}}^{{2}}$$$.

It is not difficult to discover an antiderivative of $$${{x}}^{{2}}$$$ if you know derivative of power function.

In fact, if $$${F}{\left({x}\right)}=\frac{{1}}{{3}}{{x}}^{{3}}$$$, then $$${F}'{\left({x}\right)}={{x}}^{{2}}={f{{\left({x}\right)}}}$$$.

But the function $$${G}{\left({x}\right)}=\frac{{1}}{{3}}{{x}}^{{3}}+{1}$$$ also satisfies $$${G}'{\left({x}\right)}={{x}}^{{2}}={f{{\left({x}\right)}}}$$$.

Therefore, both $$${F}$$$ and $$${G}$$$ are antiderivatives of $$${f{}}$$$.

In general, any function of the form $$$\frac{{1}}{{3}}{{x}}^{{3}}+{C}$$$, where $$${C}$$$ is a constant, is an antiderivative of $$${{x}}^{{2}}$$$.

The following fact says that $$${f{}}$$$ has no other antiderivatives.

Fact. If $$${F}{\left({x}\right)}$$$ is antiderivative of $$${f{{\left({x}\right)}}}$$$ on $$${I}$$$ then the most general antiderivative of $$${f{}}$$$ on $$${I}$$$ is $$${F}{\left({x}\right)}+{C}$$$, where $$${C}$$$ is arbitrary constant.

Example 2. Find general antiderivative of $$${{x}}^{{n}}$$$, $$${n}\ne-{1}$$$.

Since $$${\left(\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}\right)}'={{x}}^{{n}}$$$ then the general antiderivative of $$${{x}}^{{n}}$$$ is $$$\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}+{C}$$$, $$${n}\ne-{1}$$$.

In fact antiderivative arises so often that it is given special name and notation.

Definition. Expression $$${F}{\left({x}\right)}+{C}$$$, where $$${C}$$$ is arbitrary constant, that is the most general antiderivative of function $$${f{{\left({x}\right)}}}$$$ is called indefinite integral of $$${f{{\left({x}\right)}}}$$$ and is denoted by $$$\int{f{{\left({x}\right)}}}{d}{x}$$$.

In other words $$$\int{f{{\left({x}\right)}}}{d}{x}={F}{\left({x}\right)}+{C}$$$ where $$${C}$$$ is arbitrary constant.

Sign $$$\int$$$ was introduced by Leibniz and is called integral sign. $$${f{{\left({x}\right)}}}$$$ is called integrand.

Process of finding indefinite integral (or antiderivative) is called integration.
Also, bear in mind that some functions can't be integrated, in other words integral of some functions can't be done in terms of functions we know. Examples of such functions are $$$\int\frac{{{{e}}^{{x}}}}{{x}}{d}{x}$$$, $$$\int{\cos{{\left({{x}}^{{2}}\right)}}}{d}{x}$$$ etc.

Now, returning to position, velocity and acceleration function we can write that

  • $$${s}{\left({t}\right)}=\int{v}{\left({t}\right)}{d}{t}$$$
  • $$${v}{\left({t}\right)}=\int{a}{\left({t}\right)}{d}{t}$$$

Example 3. Find $$$\int{\cos{{\left({x}\right)}}}{d}{x}$$$.

Since $$${\left({\sin{{\left({x}\right)}}}\right)}'={\cos{{\left({x}\right)}}}$$$ then antiderivative (indefinite integral) of $$${\cos{{\left({x}\right)}}}$$$ is $$${\sin{{\left({x}\right)}}}$$$, so

$$$\int{\cos{{\left({x}\right)}}}{d}{x}={\sin{{\left({x}\right)}}}+{C}$$$ where $$${C}$$$ is arbitrary constant.

Since indefinite integral (general antiderivative) is given by family of functions it is impossible to define some particular function without defining some additional conditions. To define particular function we can add condition of the form $$${F}{\left({a}\right)}={b}$$$.

Example 4. Find $$$\int{{x}}^{{4}}{d}{x}$$$ if $$${F}{\left({1}\right)}={2}$$$.

Since $$${\left(\frac{{1}}{{5}}{{x}}^{{5}}\right)}'={{x}}^{{4}}$$$ then $$$\int{{x}}^{{4}}{d}{x}=\frac{{1}}{{5}}{{x}}^{{5}}+{C}$$$.

That is $$${F}{\left({x}\right)}=\frac{{1}}{{5}}{{x}}^{{5}}+{C}$$$. To define particular function (i.e. cosntant $$${C}$$$) we use condition $$${F}{\left({1}\right)}={2}$$$:

$$${2}={F}{\left({1}\right)}=\frac{{1}}{{5}}\cdot{{1}}^{{5}}+{C}$$$. From this $$${C}=\frac{{9}}{{5}}$$$.

So, $$${F}{\left({x}\right)}=\frac{{1}}{{5}}{{x}}^{{5}}+\frac{{9}}{{5}}$$$.

Now, let's study geometric meaning of indefinite integral.

Since derivative is a slope of tangent line at point $$${a}$$$ then indefinite integral is a function $$${F}{\left({x}\right)}$$$ tangent line to which at point $$${a}$$$ equals $$${f{{\left({a}\right)}}}$$$.

Example 5. Given that $$${f{{\left({x}\right)}}}=\sqrt{{{1}+{{x}}^{{2}}}}$$$. Use it to sketch graph of the indefinite integral $$${F}{\left({x}\right)}.$$$

indefinite integralRecall that slope of tangent line to the indefinite integral at point $$${a}$$$ is $$${f{{\left({a}\right)}}}$$$.

So, we calculate that $$${f{{\left({0}\right)}}}=\sqrt{{{1}+{{0}}^{{2}}}}={1}$$$, $$${f{{\left({2}\right)}}}=\sqrt{{{1}+{{2}}^{{2}}}}=\sqrt{{{5}}}\approx{2.23}$$$ etc.

This means that slope of tangent line at point 0 will be 1, slope of tangent line at point 1 will be approximately 2.23. Now we draw several tangent segments with slope 1 at point $$${x}={0}$$$ and several tangent segments with slope 2.23 at $$${x}={2}$$$. Continuing this process we obtain slopes of tangents at all points. This is called a direction field because it shows direction of $$${F}{\left({x}\right)}$$$ at all points.

Now we can draw several curves that follow direction of tangent segments. If we want particular curve then we need additional condition. For example, only green curve satisfies condition $$${F}{\left({0}\right)}={2}$$$.

Note that all curves can be obtained from each other by shifting upward or downward.