medium difficulty7 min

Properties of Definite Integrals

Properties of definite integrals transform the limits or integrand to simplify evaluation without first finding a complicated antiderivative.

In one line: Limits give the strategy: reversed means sign change, symmetric means check odd or even, 0 to a means try a minus x.

Practice This Concept

Understand it

These properties are valid under integrability conditions on the given interval. They are especially powerful for symmetry, interval splitting, reversing limits, and replacing x by a-x on [0,a]. They must be applied with the correct limits and sign.

Example

∫₀^a f(x) dx=∫₀^a f(a-x) dx, provided f is integrable on [0,a].

Formula and rule

Important properties: ∫_a^b f(x)dx=-∫_b^a f(x)dx; ∫_a^b f(x)dx=∫_a^c f(x)dx+∫_c^b f(x)dx; ∫₀^a f(x)dx=∫₀^a f(a-x)dx; ∫_-a^a f(x)dx=0 if f is odd; ∫_-a^a f(x)dx=2∫₀^a f(x)dx if f is even.

Conditions: f must be integrable on the relevant interval.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Evaluate ∫₀^π x sinx dx using the property I=∫₀^a f(x)dx=∫₀^a f(a-x)dx.

1Let I=∫₀^π x sinx dx.
2Also I=∫₀^π (π-x)sin(π-x) dx=∫₀^π (π-x)sinx dx.
3Adding, 2I=∫₀^π π sinx dx=π[-cosx]₀^π=2π.
4Hence I=π.

Method you can reuse

1Observe the limits first.
2If limits are reversed, adjust the sign.
3If the interval is split, use additivity.
4For 0 to a, try x replaced by a-x.
5For -a to a, test f(-x).
6Apply the property only after confirming integrability and simplifying correctly.

Visual tip

A graph can help with even-odd symmetry. If drawn, label the y-axis as the symmetry line for even functions and show opposite signed areas for odd functions.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is ∫_-2² x³ dx?

Think first, then show answer

0, because x³ is odd and the interval is symmetric about 0.

Common wrong answer

For an odd function on [-a,a], giving 2∫₀^a f(x)dx instead of 0 by confusing odd and even symmetry rules.

Don't mix this up

Students often reverse limits without changing the sign, writing ∫_a^b f(x)dx=∫_b^a f(x)dx, which is false unless the integral is zero.

Exam tip

For limits 0 to a, try replacing x by a-x. For limits -a to a, check whether the function is even or odd.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Often appears in higher-mark definite integral questions where direct integration is lengthy. The examiner may expect symmetry, interval transformation, or even-odd reasoning.

1
Enough for 1 mark
Properties of definite integrals transform the limits or integrand to simplify evaluation without first finding a complicated antiderivative.
2
Add for 2 marks
Example: ∫₀^a f(x) dx=∫₀^a f(a-x) dx, provided f is integrable on [0,a].
3
Add for 3 marks
These properties are valid under integrability conditions on the given interval.

MCQ Quiz

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