If f is continuous on [a,b] and F =f: _abf(x), dx=[F(x)]_ab=F(b)-F(a) Substitute into the antiderivative, upper limit first. No integration constant is needed; it cancels. Example Evaluate _13(x2+2),dx. _13(x2+2),dx =[ x33+2x]_13 =(333+2(3))-(133+2(1)) =15- 73 = 383 Check that the integrand is defined throughout the stated interval before evaluating.

If \(f\) is continuous on \([a,b]\) and \(\mathrm{F}^{\prime}=f\):

\(\int_a^bf(x)\,\mathrm dx=[\mathrm{F}(x)]_a^b=\mathrm{F}(b)-\mathrm{F}(a)\)

Substitute into the antiderivative, upper limit first. No integration constant is needed; it cancels.

Evaluate \(\displaystyle\int_1^3(x^2+2)\,\mathrm{d}x\).

\(\int_1^3(x^2+2)\,\mathrm{d}x\)\({}=\left[\dfrac{x^3}{3}+2x\right]_1^3\)\({}=\left(\dfrac{3^3}{3}+2(3)\right)-\left(\dfrac{1^3}{3}+2(1)\right)\)\({}=15-\dfrac73\)\({}=\dfrac{38}{3}\)

Check that the integrand is defined throughout the stated interval before evaluating.

Evaluating Definite Integrals

Evaluating Definite Integrals