Antiderivatives and Initial Value Problems
Definition of Antiderivative
If F (x) and f (x) are functions defined on an interval x and
for all x ∈ I, then F (x) is called an antiderivative of f (x).
Finding an antiderivative is the reverse operation to differentiation.
For example, the antiderivative of x² is \({\frac{{{x^3}}}{3}}\) because
Note that the functions \(\frac{{{x^3}}}{3} + 5,\) \(\frac{{{x^3}}}{3} - 2\) and any function \(\frac{{{x^3}}}{3} + C\) are also antiderivatives of \({x^2}\) because
Hence, if \(F\left( x \right)\) is an antiderivative of \(f\left( x \right)\) on an interval \(I,\) then the most general antiderivative of \(f\left( x \right)\) on \(I\) is
where \(C\) is an arbitrary constant.
Initial Value Problems
Finding an antiderivative of \(f\left( x \right)\) is equivalent to solving differential equation
A differential equation with an initial condition \(y\left( {{x_0}} \right) = {y_0}\) is called an initial value problem.
The most general antiderivative \(F\left( x \right) + C\) of the function \(f\left( x \right)\) gives the general solution of the differential equation \(\frac{{dy}}{{dx}} = f\left( x \right).\)
The particular solution of the initial value problem is a function that satisfies both the differential equation and the initial condition. To find the particular solution, we must apply the initial condition and determine the constant \(C.\)
Solved Problems
Example 1.
Find an antiderivative of the function \[f\left( x \right) = \frac{1}{{{x^4}}}.\]
Solution.
Notice that the derivative of the function \({\frac{1}{{{x^3}}}}\) is given by
Hence, an antiderivative has the form
We can check this by differentiation:
Example 2.
Find an antiderivative of the function \[f\left( x \right) = {e^{2x}}.\]
Solution.
It is easy to see that
So an antiderivative is given by
We can check this by differentiation:
Example 3.
Find an antiderivative of the function \[f\left( x \right) = \frac{1}{{\sqrt[3]{x}}}.\]
Solution.
Notice that
Clearly that an antiderivative is written as
because
Example 4.
Find an antiderivative of the function \[f\left( x \right) = {3^{ - x}}.\]
Solution.
It is known that
Hence
Therefore an antiderivative is given by
We can check the result by differentiation:
Example 5.
Solve the initial value problem \[\frac{{dy}}{{dx}} = {x^2} - 1, y\left( 3 \right) = 7.\]
Solution.
The general antiderivative of \({x^2} - 1\) is
Substitute the initial condition \(y\left( 3 \right) = 7\) to determine the value of \(C:\)
Hence, the solution of the initial value problem is given by
Example 6.
Solve the initial value problem \[\frac{{dy}}{{dx}} = 2x - \frac{1}{{{x^2}}}, x \ne 0, y\left( 1 \right) = 5.\]
Solution.
First we write the general solution (general antiderivative) of the differential equation. As
then
Determine the constant \(C\) using the initial condition \(y\left( 1 \right) = 5:\)
Hence, the particular solution of the initial value problem has the form