For a continuous function on a closed interval, evaluate an antiderivative at the upper and lower limits. A definite integral is a number; it has no arbitrary constant. _ab f(x),dx=[F(x)]_ab=F(b)-F(a). _13(2x+1),dx=[x2+x]_13=12-2=10. **Property** | **Identity** Reversed limits | _ba f(x),dx=- _ab f(x),dx Split an interval | _ab f(x),dx= _ac f(x),dx+ _cb f(x),dx, a<c<b Constant multiple | _ab kf(x),dx=k _ab f(x),dx With u=x2+1, du=2x,dx, the limits x=0,1 become u=1,2: _01 2x(x2+1)3,dx= _12u3,du=[ u44]_12= 154. Use limits in the same variable as the integrand. Alternatively substitute back to the original variable before using the original limits. When the relevant integrals exist, _aa f(x),dx=0, and sums can be integrated term by term. The additivity and reversed-limit rules follow by cancelling the matching antiderivative endpoint values. Do not apply endpoint subtraction across an undefined or unbounded point without checking existence.

Evaluating Definite Integrals

IB Year 6 | Grade 12

IB AA HL integration notes — 23 October 2026

For a continuous function on a closed interval, evaluate an antiderivative at the upper and lower limits. A definite integral is a number; it has no arbitrary constant.

\(\int_a^b \mathrm{f}(x)\,\mathrm{d}x=[\mathrm{F}(x)]_a^b=\mathrm{F}(b)-\mathrm{F}(a)\).

Evaluation

\(\int_1^3(2x+1)\,\mathrm{d}x=[x^2+x]_1^3=12-2=10\).

Property	Identity
Reversed limits	\(\int_b^a \mathrm{f}(x)\,\mathrm{d}x=-\int_a^b \mathrm{f}(x)\,\mathrm{d}x\)
Split an interval	\(\int_a^b \mathrm{f}(x)\,\mathrm{d}x=\int_a^c \mathrm{f}(x)\,\mathrm{d}x+\int_c^b \mathrm{f}(x)\,\mathrm{d}x\), \(a<c<b\)
Constant multiple	\(\int_a^b k\mathrm{f}(x)\,\mathrm{d}x=k\int_a^b \mathrm{f}(x)\,\mathrm{d}x\)

Changing limits after substitution

With \(u=x^2+1\), \(\mathrm{d}u=2x\,\mathrm{d}x\), the limits \(x=0,1\) become \(u=1,2\):

\(\int_0^1 2x(x^2+1)^3\,\mathrm{d}x=\int_1^2u^3\,\mathrm{d}u=\left[\dfrac{u^4}{4}\right]_1^2=\dfrac{15}{4}\).

Use limits in the same variable as the integrand. Alternatively substitute back to the original variable before using the original limits.

When the relevant integrals exist, \(\int_a^a f(x)\,\mathrm{d}x=0\), and sums can be integrated term by term. The additivity and reversed-limit rules follow by cancelling the matching antiderivative endpoint values. Do not apply endpoint subtraction across an undefined or unbounded point without checking existence.

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