H2 Math Integration by Parts: Choose u and Understand the Formula
With Mr Gan13 min 30 secPublished
Watch Mr Gan derive integration by parts from the product rule, explain how to choose u and work through the integral of x ln x for A-Level H2 Mathematics.
What you will learn
- Connect integration by parts to the product rule.
- Choose u so that the remaining integral becomes easier.
- Set out u, du/dx, dv/dx and v, then check the answer by differentiation.
Lesson notes
A written summary of the method, with mathematical notation clarified for revision.
Why the formula works
Start from the product rule: d(uv)/dx = u(dv/dx) + v(du/dx). Rearranging and integrating gives ∫u dv = uv − ∫v du. The aim is to replace the original integral with one that is easier to evaluate.
Choose u with the next step in mind. Logarithmic terms are often useful choices because differentiating them produces a simpler expression. LIATE is a guideline, rather than a rule that works for every integral.
Worked example: integrate x ln x
For x > 0, let u = ln x and dv/dx = x. Then du/dx = 1/x and v = x²/2.
Substitute into the formula: ∫x ln x dx = (x²/2) ln x − ∫x/2 dx = (x²/2) ln x − x²/4 + C.
Differentiate the final expression. The extra x/2 terms cancel, leaving x ln x. This checks both the algebra and the sign in the integration-by-parts formula.
Try it yourself
Try integrating x² ln x for x > 0. Choose u before calculating, and differentiate your answer to check it.
Check your reasoning
Let u = ln x and dv/dx = x². The result is (x³/3) ln x − x³/9 + C.