Integration by Parts for A/L: When to Use It and How to Choose u

Integration by parts is one of the most tested techniques in A/L pure mathematics. The formula is easy to remember; choosing the parts well is the real skill.

The formula

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

It comes straight from the product rule for differentiation, which is a good way to remember it if you ever forget the signs.

When to use it

Use integration by parts when the integrand is a product of two different types of function, for example:

• $x e^{x}$ — a polynomial times an exponential
• $x \sin x$ — a polynomial times a trigonometric function
• $\ln x$ — which you can treat as $1 \cdot \ln x$

How to choose u

Choose $u$ to be the part that gets simpler when you differentiate it. A useful order of preference (often remembered as LIATE) is:

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

Whichever appears first in the list is usually your $u$.

Worked example 1

Find $\displaystyle \int x e^{x} \, dx$.

Let $u = x$ and $\dfrac{dv}{dx} = e^{x}$. Then $\dfrac{du}{dx} = 1$ and $v = e^{x}$.

$\int x e^{x} \, dx = x e^{x} - \int e^{x} \, dx = x e^{x} - e^{x} + C$

Worked example 2

Find $\displaystyle \int \ln x \, dx$.

Write it as $\int 1 \cdot \ln x \, dx$. Let $u = \ln x$ and $\dfrac{dv}{dx} = 1$, so $\dfrac{du}{dx} = \dfrac{1}{x}$ and $v = x$.

$\int \ln x \, dx = x \ln x - \int x \cdot \frac{1}{x} \, dx = x \ln x - x + C$

Common mistakes

• Choosing u the wrong way round. If the new integral is harder than the original, swap your choice.
• Sign errors in the minus between the two terms — write the formula out every time.
• Forgetting $+ C$ in indefinite integrals, or forgetting to apply the limits to the $uv$ term in definite integrals.
• Stopping too early. Some questions, like $\int x^2 e^{x} \, dx$, need integration by parts twice.

Practise

Work through the integration questions from the last five years of past papers. If you keep getting stuck at the same step, a Math Power Slot in the MathCLINIC will find out why.