A definite integral is evaluated from an antiderivative; the upper-limit value is taken minus the lower-limit value. Fundamental theorem of calculus \[\int_a^b \mathrm{f}(x)\,\mathrm{d}x=\bigl[\mathrm{F}(x)\bigr]_a^b=\mathrm{F}(b)-\mathrm{F}(a),\qquad \mathrm{F}'(x)=\mathrm{f}(x)\] Useful properties PropertyResult Equal limits\(\displaystyle\int_a^a \mathrm{f}(x)\,\mathrm{d}x=0\) Reversed limits\(\displaystyle\int_a^b \mathrm{f}(x)\,\mathrm{d}x=-\int_b^a \mathrm{f}(x)\,\mathrm{d}x\) Adjacent intervals\(\displaystyle\int_a^b \mathrm{f}(x)\,\mathrm{d}x+\int_b^c \mathrm{f}(x)\,\mathrm{d}x=\int_a^c \mathrm{f}(x)\,\mathrm{d}x\) Constant multiple\(\displaystyle\int_a^b k\mathrm{f}(x)\,\mathrm{d}x=k\int_a^b \mathrm{f}(x)\,\mathrm{d}x\) Sum or difference\(\displaystyle\int_a^b[\mathrm{f}(x)\pm \mathrm{g}(x)]\,\mathrm{d}x=\int_a^b \mathrm{f}(x)\,\mathrm{d}x\pm\int_a^b \mathrm{g}(x)\,\mathrm{d}x\) Change of variables If \(u=\mathrm{g}(x)\), transform both limits as well as the integrand. When \(x=a\), \(u=\mathrm{g}(a)\); when \(x=b\), \(u=\mathrm{g}(b)\). Substitution with transformed limits \[\int_a^b \mathrm{f}\bigl(g(x)\bigr)g'(x)\,\mathrm{d}x=\int_{g(a)}^{g(b)}\mathrm{f}(u)\,\mathrm{d}u\] After changing the limits, evaluate entirely in the new variable; there is no need to substitute back.

Definite Integrals and Substitution

Junior College 1

A definite integral is evaluated from an antiderivative; the upper-limit value is taken minus the lower-limit value.

Fundamental theorem of calculus

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

Useful properties

Property	Result
Equal limits	\(\displaystyle\int_a^a \mathrm{f}(x)\,\mathrm{d}x=0\)
Reversed limits	\(\displaystyle\int_a^b \mathrm{f}(x)\,\mathrm{d}x=-\int_b^a \mathrm{f}(x)\,\mathrm{d}x\)
Adjacent intervals	\(\displaystyle\int_a^b \mathrm{f}(x)\,\mathrm{d}x+\int_b^c \mathrm{f}(x)\,\mathrm{d}x=\int_a^c \mathrm{f}(x)\,\mathrm{d}x\)
Constant multiple	\(\displaystyle\int_a^b k\mathrm{f}(x)\,\mathrm{d}x=k\int_a^b \mathrm{f}(x)\,\mathrm{d}x\)
Sum or difference	\(\displaystyle\int_a^b[\mathrm{f}(x)\pm \mathrm{g}(x)]\,\mathrm{d}x=\int_a^b \mathrm{f}(x)\,\mathrm{d}x\pm\int_a^b \mathrm{g}(x)\,\mathrm{d}x\)

Change of variables

If \(u=\mathrm{g}(x)\), transform both limits as well as the integrand. When \(x=a\), \(u=\mathrm{g}(a)\); when \(x=b\), \(u=\mathrm{g}(b)\).

Substitution with transformed limits

\[\int_a^b \mathrm{f}\bigl(g(x)\bigr)g'(x)\,\mathrm{d}x=\int_{g(a)}^{g(b)}\mathrm{f}(u)\,\mathrm{d}u\]

After changing the limits, evaluate entirely in the new variable; there is no need to substitute back.

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