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Some definite integrals look impossible. Try to find an antiderivative of sin x / (sin x + cos x) and you will fill a page. Yet the definite integral from 0 to π/2 takes three lines. JEE uses such integrals to test whether you know the properties, not whether you can integrate by brute force. This post works through the properties that matter most, with full worked examples and the conditions that must hold before you use each one.
What does the syllabus include?
The JEE (Main) 2026 syllabus lists "the fundamental theorem of calculus, properties of definite integrals" and "evaluation of definite integrals" under Integral Calculus. The NCERT Class 12 chapter on integrals states and proves the properties as P0 to P7. Check the syllabus for your exam year.
Why think of an integral as a sum?
The properties make sense once you see a definite integral as adding up many small pieces, f(x) dx, across an interval. Research suggests this view is both hard to build and worth having. A. Orton (1983) interviewed 100 students aged 16 to 22 and found that items on integration as the limit of a sum were difficult even for good students. Steven Jones (2013) found that thinking of the integral as an addition over many pieces was especially useful in multivariable and physics contexts. Keep that picture as you read each property: most of them say that the same pieces can be added in a different order.
Which properties do you need?
From NCERT, in the form you will use most:
P1: swapping the limits changes the sign.
P2: you can split at any point c: ∫ from a to b = ∫ from a to c + ∫ from c to b.
P3 (the "King's property"): ∫ from a to b of f(x) dx = ∫ from a to b of f(a + b - x) dx.
P4: the special case ∫ from 0 to a of f(x) dx = ∫ from 0 to a of f(a - x) dx.
P6: ∫ from 0 to 2a of f(x) dx = 2 ∫ from 0 to a of f(x) dx if f(2a - x) = f(x), and 0 if f(2a - x) = -f(x).
P7: over [-a, a], an even function gives 2 ∫ from 0 to a, and an odd function gives 0.
Worked example 1: the classic P4 integral
Find I = ∫ from 0 to π/2 of sin x / (sin x + cos x) dx.
By P4, replace x with π/2 - x. Since sin(π/2 - x) = cos x and cos(π/2 - x) = sin x, I = ∫ from 0 to π/2 of cos x / (cos x + sin x) dx.
Add the two forms: 2I = ∫ from 0 to π/2 of (sin x + cos x)/(sin x + cos x) dx = ∫ from 0 to π/2 of 1 dx = π/2.
So I = π/4.
Worked example 2: P3 with general limits
Find I = ∫ from 2 to 8 of √x / (√x + √(10 - x)) dx.
Here a + b = 10, so P3 replaces x with 10 - x: I = ∫ from 2 to 8 of √(10 - x) / (√(10 - x) + √x) dx.
Add: 2I = ∫ from 2 to 8 of 1 dx = 6.
So I = 3.
The pattern to spot: the integrand and its "a + b - x" version add to something simple.
Worked example 3: when x appears outside
Find I = ∫ from 0 to π of x sin x / (1 + cos² x) dx.
By P4, replace x with π - x. Since sin(π - x) = sin x and cos²(π - x) = cos² x, I = ∫ from 0 to π of (π - x) sin x / (1 + cos² x) dx.
Add the two forms: 2I = π ∫ from 0 to π of sin x / (1 + cos² x) dx.
Substitute t = cos x, dt = -sin x dx. When x = 0, t = 1; when x = π, t = -1. The integral becomes ∫ from -1 to 1 of dt / (1 + t²) = tan⁻¹(1) - tan⁻¹(-1) = π/4 + π/4 = π/2.
So 2I = π × π/2 = π²/2, and I = π²/4.
Worked example 4: odd, even and P6
∫ from -2 to 2 of (x³ + x cos x) dx. Both x³ and x cos x are odd, since replacing x with -x flips the sign. By P7, the integral is 0.
∫ from 0 to π of |cos x| dx. Here |cos(π - x)| = |cos x|, so by P6 the integral is 2 ∫ from 0 to π/2 of cos x dx = 2 × 1 = 2.
Check odd or even by substituting -x, not by looking. x sin x is even, because both factors flip sign. x² sin x is odd.
Common slips and how to avoid them
Using P7 on an interval that is not symmetric about zero.
Calling a function odd without testing f(-x).
Missing an x when substituting a + b - x.
Using P4's form with limits that do not start at 0. Use P3 instead.
Forgetting to change the limits after a substitution, as in step 3 of example 3.
A practice routine
Spot the property: for ten integrals, write only which property you would try and why. Then solve.
King's property set: five integrals of the form f(x) / (f(x) + f(a + b - x)).
Odd and even: five integrals over [-a, a] where you must split the integrand into odd and even parts.
Mixed timed set from past papers, checking each answer numerically with a calculator where you can.
Once these feel routine, the same ideas power area under curves. If an earlier step such as substitution is shaky, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Today, solve ∫ from 0 to π/2 of cos x / (sin x + cos x) dx using P4, and check that it matches example 1.
