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CIE A-Level Maths Study Notes

1.8.2 Finding the Constant of Integration

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)F(x) is an antiderivative of f(x)f(x), the most general antiderivative of f(x)f(x) is F(x)+CF(x) + C, where CC 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))(1, -2) ), these are used to solve for 'C'.

Solving a Differential Equation

Example 1

Given: Differential equation dydx=x34x+1\frac{dy}{dx} = x^3 - 4x + 1, with the condition that the solution passes through (0,2)(0, 2).


1. Integrate x34x+1x^3 - 4x + 1 with respect to xx:

y=(x34x+1)dxy = \int (x^3 - 4x + 1) \, dx

2. Add the constant of integration CC:

y=x442x2+x+Cy = \frac{x^4}{4} - 2x^2 + x + C

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

2=0442(0)2+0+CC=22 = \frac{0^4}{4} - 2(0)^2 + 0 + C \Rightarrow C = 2

4. The particular solution is:

y=x442x2+x+2y = \frac{x^4}{4} - 2x^2 + x + 2

Example 2

Given: f(x)=sin(x)+cos(x)f(x) = \sin(x) + \cos(x), with the curve passing through (π2,1)\left( \frac{\pi}{2}, 1 \right).


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

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

2. The integral yields:

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

3. Use (π2,1)\left( \frac{\pi}{2}, 1 \right) to find CC:

1=cos(π2)+sin(π2)+CC=01 = -\cos\left(\frac{\pi}{2}\right) + \sin\left(\frac{\pi}{2}\right) + C \Rightarrow C = 0

4. Thus, the antiderivative with the constant is:

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

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

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.

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