The following rules require all relevant integrals to exist. \(\int_a^a\mathrm f(x)\,\mathrm dx=0\qq

Question

The following rules require all relevant integrals to exist.

$\int_a^a\mathrm f(x)\,\mathrm dx=0\qquad\int_a^b\mathrm f(x)\,\mathrm dx=-\int_b^a\mathrm f(x)\,\mathrm dx$

$\int_a^c\mathrm f(x)\,\mathrm dx=\int_a^b\mathrm f(x)\,\mathrm dx+\int_b^c\mathrm f(x)\,\mathrm dx$

Reversing the limits reverses the subtraction: $\mathrm{F}(a)-\mathrm{F}(b)=-[\mathrm{F}(b)-\mathrm{F}(a)]$.

When the interval is split at $b$, the two contributions add because $[\mathrm{F}(b)-\mathrm{F}(a)]+[\mathrm{F}(c)-\mathrm{F}(b)]=\mathrm{F}(c)-\mathrm{F}(a)$.

Constant multiples and sums can be integrated term by term. Check undefined or unbounded points before applying these rules.

Accepted Answer

$\int_b^a \mathrm f(x)\,\mathrm dx=-\int_a^b \mathrm f(x)\,\mathrm dx$ $\int_a^c \mathrm f(x)\,\mathrm dx=\int_a^b \mathrm f(x)\,\mathrm dx+\int_b^c \mathrm f(x)\,\mathrm dx$

Understanding Results of Definite Integrals

Understanding Results of Definite Integrals

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