medium difficulty6 min

Definite Integrals and the Fundamental Theorem of Calculus

If F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx=F(b)-F(a). This result is the evaluation form of the Fundamental Theorem of Calculus.

In one line: Upper value minus lower value; the constant cancels.

Practice This Concept

Understand it

A definite integral gives a number, not a family of functions. The limits fix the interval, so no arbitrary constant is written in the final value. The function should be integrable on the interval, and the antiderivative used must be valid throughout the interval.

Example

∫₀² x² dx=[x³/3]₀²=8/3-0=8/3.

Formula and rule

FTC evaluation rule: ∫_a^b f(x) dx=F(b)-F(a), where F'(x)=f(x) and f is integrable on [a,b].

No +C appears in the final definite integral value because the constant cancels.

See it worked out

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

Worked example

Evaluate ∫₀^π sinx dx.

1Antiderivative of sinx is -cosx.
2Therefore ∫₀^π sinx dx=[-cosx]₀^π=-cosπ-(-cos0)=1-(-1)=2.
3Final value: 2.

Method you can reuse

1Check the limits and integrand.
2Find a valid antiderivative.
3Write [F(x)] from a to b.
4Substitute the upper limit first and lower limit second.
5Simplify F(b)-F(a).
6Do not add +C.

Visual tip

A graph may help interpret signed area, but routine FTC evaluation does not require a diagram. If used, label x=a, x=b, y=f(x), and signed regions above or below the x-axis.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Evaluate ∫₁³ 2x dx.

Think first, then show answer

[x²]₁³=9-1=8.

Common wrong answer

For ∫₀^π sinx dx, writing cosπ-cos0=-2 because the antiderivative of sinx was taken as cosx instead of -cosx.

Don't mix this up

Students sometimes add +C in a definite integral or substitute limits in the wrong order as F(a)-F(b).

Exam tip

Always write the antiderivative in bracket notation before substituting limits; it reduces sign and order errors.

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

Asked as direct definite integral evaluation, as part of area/accumulation interpretation, or after using substitution, identities, partial fractions, or by-parts to find the antiderivative.

1
Enough for 1 mark
If F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx=F(b)-F(a). This result is the evaluation form of the Fundamental Theorem of Calculus.
2
Add for 2 marks
A definite integral gives a number, not a family of functions.
3
Add for 3 marks
The limits fix the interval, so no arbitrary constant is written in the final value.

MCQ Quiz

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