Definite Integral
Definite Integral
A definite integral carries limits and evaluates to a number. If \(\displaystyle\int f(x)\,\mathrm{d}x=F(x)+c\), then \(\displaystyle\int_a^b f(x)\,\mathrm{d}x=\big[F(x)\big]_a^b=F(b)-F(a)\).
Evaluating a definite integral
| Step | Working |
|---|---|
| Integrate; write the result in square brackets with the limits | |
| Substitute the upper limit, then subtract the lower |
No \(+c\) is needed for a definite integral: it would appear in both brackets and cancel in the subtraction.
Exam check
Substitute the limits into the integrated expression, never into the original integrand, and keep the order: upper limit first, then subtract the lower.
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Answer:\(\int_b^a f(x)\,\mathrm dx=F(a)-F(b)\) Example: \(\int_1^3(x^2+2)\,\mathrm dx=\frac{38}{3}\)
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