Definite integrals: five properties beat computation

A large share of JEE definite-integral questions are not asking you to integrate. They are asking whether you recognise a property that removes the need to. If you find yourself doing heavy algebra on a definite integral, you have usually missed the intended move.

The five properties worth knowing cold

Reflection. For any $f$ on the interval from $a$ to $b$: $$\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$$ Adding the original and the reflected form often collapses the integrand to a constant. This is the workhorse: more JEE questions yield to this one property than to any other. Symmetry about zero. Over a symmetric interval, an odd integrand gives zero and an even one gives twice the half-integral. Check parity before doing anything — it takes three seconds and sometimes finishes the question. Periodicity. If $f$ has period $T$, the integral over $n$ full periods is $n$ times the integral over one, and the starting point does not matter. Questions with limits like $0$ to $100\pi$ exist to test this. The half-period swap. On the interval from $0$ to $\pi/2$, substituting $x \to \pi/2 - x$ exchanges sine and cosine. Any integrand built symmetrically from the two is immediately halved. Splitting at a kink. Integrands containing a modulus, a greatest-integer function or a piecewise definition must be split where the behaviour changes. Most errors on these come from not splitting, not from the integration.

Recognising which one applies

A checklist, run before you touch the integrand:

1. Are the limits symmetric about zero? Check parity.
2. Are the limits $0$ to $\pi/2$, or $0$ to $\pi$? Try the reflection substitution.
3. Is the integrand periodic and the interval long? Reduce to one period.
4. Is there a modulus, a floor function or a sign change inside? Split.
5. Do the limits sum to something clean? Reflection again.

Running that list costs fifteen seconds and converts a fifteen-minute question into a three-minute one often enough to move your score.

The classic worth having seen

$$\int_0^{\pi/2} \frac{dx}{1 + \tan^n x} = \frac{\pi}{4}$$ for every $n$. The reflection substitution produces a second integral whose integrand is the complement of the first, so the two add to the length of the interval and each is half of it. Any question with a comparable structure — some function of tangent, sine or cosine sitting in a denominator with a 1 added — is a candidate for the same trick.

Practise properties, not integration

Take twenty definite integrals from previous papers and, without solving any, write down which property you would use. Then check. This separates recognition from computation, and recognition is the skill being tested.