Integration is a key component of calculus, essential for solving various problems in fields like physics and engineering. In this section, the focus is on the concept of the constant of integration and exploring methods to determine this constant from given conditions, applying it to particular solutions of differential equations.

Mathematical Representation

Mathematically, if $F(x)$ is an antiderivative of $f(x)$, the most general antiderivative of $f(x)$ is $F(x) + C$, where $C$ represents any real number.

Techniques for Evaluating the Constant of Integration

From General to Particular Solutions

In practical scenarios, especially in physics and engineering, it's often necessary to find a particular solution that adheres to specific initial or boundary conditions. This involves determining the value of the constant of integration, 'C'.

Using Initial Conditions

When provided with specific conditions, such as a curve passing through a known point (e.g., $(1, -2) )$, these are used to solve for 'C'.

Solving a Differential Equation

Example 1

Given: Differential equation $\frac{dy}{dx} = x^3 - 4x + 1$, with the condition that the solution passes through $(0, 2)$.

Solution:

1. Integrate $x^3 - 4x + 1$ with respect to $x$:

$y = \int (x^3 - 4x + 1) \, dx$

2. Add the constant of integration $C$:

$y = \frac{x^4}{4} - 2x^2 + x + C$

3. Apply $y(0) = 2$ to find $C$:

$2 = \frac{0^4}{4} - 2(0)^2 + 0 + C \Rightarrow C = 2$

4. The particular solution is:

$y = \frac{x^4}{4} - 2x^2 + x + 2$

Example 2

Given: $f(x) = \sin(x) + \cos(x)$, with the curve passing through $\left( \frac{\pi}{2}, 1 \right)$.

Solution:

1. Integrate $\sin(x) + \cos(x)$ with respect to $x$:

$F(x) = \int (\sin(x) + \cos(x)) \, dx$

2. The integral yields:

$F(x) = -\cos(x) + \sin(x) + C$

3. Use $\left( \frac{\pi}{2}, 1 \right)$ to find $C$:

$1 = -\cos\left(\frac{\pi}{2}\right) + \sin\left(\frac{\pi}{2}\right) + C \Rightarrow C = 0$

4. Thus, the antiderivative with the constant is:

$F(x) = -\cos(x) + \sin (x)$