Mastering A-Level Pure Mathematics: Integration by Parts & Differential Equations

Tackling Year 2 Pure Mathematics with Confidence

Integration by parts and separable differential equations represent core pillars of the A-Level Pure Mathematics syllabus. Mastering these topics guarantees high scores in Paper 1 and Paper 2.

Understanding the Integration by Parts Formula

The standard formula derived from the product rule of differentiation is given by:

$$\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$$

Using the LIATE Rule for Selecting $u$

When faced with a product of two distinct functions, select $u$ in order of priority:

1. Logarithmic functions (e.g. $\ln x$)
2. Inverse trigonometric functions (e.g. $\arcsin x$)
3. Algebraic polynomials (e.g. $x^2, 3x$)
4. Trigonometric functions (e.g. $\sin x, \cos x$)
5. Exponential functions (e.g. $e^{2x}$)

Worked Example: Integrating $x e^{3x}$

Let $u = x \implies \frac{du}{dx} = 1$, and $\frac{dv}{dx} = e^{3x} \implies v = \frac{1}{3}e^{3x}$.

Applying the formula:

$$\int x e^{3x} dx = \frac{1}{3}x e^{3x} - \int \frac{1}{3}e^{3x} dx = \frac{1}{3}x e^{3x} - \frac{1}{9}e^{3x} + C$$

First-Order Separable Differential Equations

Whenever you encounter $\frac{dy}{dx} = f(x)g(y)$, always separate variables before integrating:

$$\int \frac{1}{g(y)} dy = \int f(x) dx$$

Never forget to include the constant of integration $+ C$ on one side immediately after evaluating the integrals, before applying initial boundary conditions.