Edexcel IAL - Pure Maths 1- 5.1 Indefinite Integration as the Reverse of Differentiation- Study notes - New syllabus
Edexcel IAL – Pure Maths 1- 5.1 Indefinite Integration as the Reverse of Differentiation -Study notes- New syllabus
Edexcel IAL – Pure Maths 1- 5.1 Indefinite Integration as the Reverse of Differentiation -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- Indefinite Integration as the Reverse of Differentiation
Indefinite Integration as the Reverse of Differentiation
Indefinite integration is the process of finding a function whose derivative is a given expression. Since differentiation removes constants, integration introduces an arbitrary constant.
Key Idea
If \( \dfrac{d}{dx}(F(x)) = f(x) \), then the indefinite integral is \( \displaystyle \int f(x)\,dx = F(x) + C \)
Here, \( C \) is the constant of integration, required whenever the integral is indefinite.
Why is the constant required?
- Because many functions differ only by a constant but share the same derivative
- E.g., the derivative of \( x^2 + 5 \), \( x^2 – 3 \), and \( x^2 + 100 \) is always \( 2x \)
- Thus \( \displaystyle \int 2x\,dx = x^2 + C \)
Basic Integration Rules (Reversing Differentiation)
| Function | Indefinite Integral |
| \( x^n \) (for \( n \ne -1 \)) | \( \displaystyle \int x^n dx = \dfrac{x^{n+1}}{n+1} + C \) |
| \( \cos x \) | \( \displaystyle \int \cos x\,dx = \sin x + C \) |
| \( \sin x \) | \( \displaystyle \int \sin x\,dx = -\cos x + C \) |
| \( e^x \) | \( \displaystyle \int e^x dx = e^x + C \) |
Example
Find \( \displaystyle \int 6x\,dx \).
▶️ Answer / Explanation
\( \int 6x\,dx = 6 \cdot \dfrac{x^2}{2} + C = 3x^2 + C \)
Example
Find \( \displaystyle \int (4x^3 – 5x)\,dx \).
▶️ Answer / Explanation
\( \int 4x^3 dx = x^4 \)
\( \int -5x dx = -\dfrac{5}{2}x^2 \)
So the result is:
\( x^4 – \dfrac{5}{2}x^2 + C \)
Example
Find \( \displaystyle \int (2e^{3x})\,dx \).
▶️ Answer / Explanation
Use reverse chain rule idea:
\( \int 2e^{3x}\,dx = 2 \cdot \dfrac{1}{3} e^{3x} + C \)
\( = \dfrac{2}{3}e^{3x} + C \)