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:

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

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.