Definite Integration (Class 12 Maths): Properties + Board-Style Solved Questions
The key properties of definite integrals and how to use them to solve common CBSE/ISC board questions quickly.
Quick summary: A definite integral gives the net area under a curve between two limits. Most board questions become easy once you use the right property — especially the “limits swap” and “a+b−x” properties.
Properties you must know
∫ab f(x) dx = −∫ba f(x) dx
∫ab f(x) dx = ∫ab f(a+b−x) dx
∫0a f(x) dx = ∫0a f(a−x) dx
The second and third are the workhorses for board problems with trigonometric integrands.
Worked example 1 — basic evaluation
Q. Evaluate ∫02 x² dx.
= [x³/3]02 = 8/3 − 0 = 8/3
Worked example 2 — using the a−x property
Q. Evaluate I = ∫0π/2 sin x /(sin x + cos x) dx.
By the property ∫0af(x)dx = ∫0af(a−x)dx, replace x with π/2−x to get I = ∫0π/2 cos x /(cos x + sin x) dx. Adding the two forms:
2I = ∫0π/2 1 dx = π/2 ⇒ I = π/4
Exam tips
- Check whether the a+b−x property removes the awkward part of the integrand.
- For even/odd functions on symmetric limits, use ∫−aa = 2∫0a (even) or 0 (odd).
- Always apply limits only after integrating.
FAQs
How much does calculus carry in Class 12 boards?
Calculus is the highest-weight unit in Class 12 Maths. Confirm the exact marks split from the current session’s CBSE/ISC syllabus.
Is there a shortcut for these integrals?
The properties above ARE the shortcuts — recognising which one to apply is the real skill.
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