Definite Integration and Its Properties
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Direct answer
The definite integral of f from a to b equals F(b) - F(a) for any antiderivative F, but JEE questions are rarely about brute antiderivatives — they are about symmetry. The King property rewrites the integral 0 to a of f(x) dx as the integral 0 to a of f(a - x) dx; odd-even and periodicity arguments collapse ugly integrands into cheap ones, and the Leibniz rule differentiates integrals with variable limits.
What you must remember
- King property: integral 0 to a of f(x) dx = integral 0 to a of f(a - x) dx; more generally, integral a to b of f(x) dx = integral a to b of f(a + b - x) dx.
- Even-odd: integral from -a to a is twice the integral from 0 to a for even f, and zero for odd f — valid only because the limits are symmetric about 0.
- Periodicity: if f has period T, the integral over any interval of length nT equals n times the integral over one period; integral 0 to nT = n × integral 0 to T.
- Classic values: integral 0 to pi/2 of dx/(1 + tan x) = pi/4; integral 0 to pi/2 of sin^n x/(sin^n x + cos^n x) dx = pi/4 for any n; integral 0 to pi/2 of ln(sin x) dx = -(pi/2) ln 2.
- Leibniz rule: d/dt of the integral from a(t) to b(t) of f(x) dx = f(b(t)) b'(t) - f(a(t)) a'(t); if the integrand also depends on t, add the integral of the partial derivative with respect to t.
- Signed area: a definite integral counts area with sign; positive and negative lobes cancel — geometric area needs |f(x)| and zeros as split points.
Common confusion
Substitution errors under the King property cost the most marks: when x is replaced by a - x, dx becomes -dx and the limits reverse, and the two minus signs must cancel visibly. Even-odd reasoning needs limits symmetric about zero — sin x over [0, pi] is handled by x → pi - x, not by oddness. And in the Leibniz rule the upper limit carries a plus and the lower a minus; reversing them is the classic slip.
Exam-focused takeaway
JEE Main asks property-based evaluation — King, even-odd, periodicity — plus simple definite integrals and Leibniz differentiation as numerical answers. JEE Advanced builds longer chains: paired substitutions that halve work, integrals defined recursively through integration by parts, inequalities between integrals, and functions defined as integrals whose monotonicity or limits must be probed. The reflex to train: before integrating, scan limits and integrand for symmetry worth money.
Frequently asked questions
What is the King property?
Integral a to b of f(x) dx equals integral a to b of f(a + b - x) dx; averaging the two forms often collapses the integrand, the standard trick for 0 to pi/2 fractions.
What is the integral of an odd function over symmetric limits?
Exactly zero, since matched intervals on either side of 0 contribute equal and opposite amounts.
What is the value of the integral of ln(sin x) from 0 to pi/2?
-(pi/2) ln 2, by pairing the integral with itself under x → pi/2 - x and solving.
When is the Leibniz rule needed?
When the integral's limits depend on a variable and you must differentiate; the upper limit brings f(b) b' with a plus sign, the lower f(a) a' with a minus.
Can a definite integral be negative?
Yes — it is signed area; parts of f below the axis subtract, and geometric area needs the modulus with splits at zeros.