Integration (Class 12 Maths): Choosing the Right Method First
Most integration marks are lost before any integrating happens — in choosing the wrong method and committing three lines to it. Recognition is the skill worth building.
A decision procedure, not a toolbox
Students who struggle with integration usually know all three methods and cannot tell which one a given problem wants. Running the same check every time fixes that faster than more practice does.
| If the integrand looks like this | Use | Why |
|---|---|---|
| A composite function with its inner derivative also present | Substitution | Reverses the chain rule |
| A product of two different function types (e.g. x·ex) | By parts, ILATE order | Reverses the product rule |
| A rational function, denominator factorisable | Partial fractions | Splits into standard forms |
| sin or cos to an even power | Trig identity first | Reduce the power before integrating |
| √(a2 − x2) or similar | Trigonometric substitution | Turns the surd into a trig function |
By parts, and what ILATE is actually for
ILATE — Inverse, Logarithmic, Algebraic, Trigonometric, Exponential — tells you which factor to call u. It is a priority order, not a law: choose as u whichever function comes first in that list, because that is the one whose derivative simplifies. The standard exam trap is ∫log x dx, which looks like a single function and is done by parts with u = log x and dv = dx. Students who have not seen that once rarely find it under pressure.
Definite integrals: the properties that save time
- ∫ab f(x) dx = ∫ab f(a + b − x) dx — the single most useful property in the chapter, and the one boards test most.
- ∫−aa f(x) dx = 0 if f is odd, and 2∫0a f(x) dx if f is even. Checking parity first can turn a long problem into one line.
- ∫02a f(x) dx = 2∫0a f(x) dx when f(2a − x) = f(x).
Before integrating any definite integral, spend ten seconds checking whether a property applies. When one does, it usually replaces most of the work.
What to practise
Not more integrals — more recognition. Take fifty problems and, without solving any of them, write only the method you would use. Check against the answers. That exercise moves marks faster than solving twenty problems slowly, because method choice is where the time and the marks are actually going.
FAQs
How do I know when to use substitution rather than by parts?
Look for a function and its own derivative both appearing in the integrand — that is substitution. By parts is for a product of two unrelated function types, where neither is the derivative of the other.
Is ILATE a rule or a guideline?
A guideline that works almost always at this level. It orders the functions by how much simpler they get on differentiation, which is what makes by parts terminate rather than loop.
Do I need to change limits when substituting in a definite integral?
Yes, unless you convert back to the original variable before applying the limits. Changing the limits is usually quicker and is the method examiners expect to see.
Which definite-integral property is most examined?
∫ab f(x) dx = ∫ab f(a + b − x) dx. It appears in some form nearly every year and converts several otherwise long problems into short ones.
Need help with this chapter?
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