2 Substitution and definite integration

If you are dealing with definite integrals (ones with limits of integration) you must be particularly careful when you substitute. Consider the following example.

Example 20

Find the definite integral ${\int \; <!--nolimits\; -->}_2^{3}tsin\left(t^2\right)dt$ by making the substitution $u=t^2$ .

Solution

Note that if $u=t^2$ then $\frac{du}{dt}=2t$ so that $dt=\frac{du}{2t}$ . We find

${\int \; <!--nolimits\; -->}_{t=2}^{t=3}tsin\left(t^2\right)dt={\int \; <!--nolimits\; -->}_{t=2}^{t=3}tsinu\frac{du}{2t}=\frac{1}{2}{\int \; <!--nolimits\; -->}_{t=2}^{t=3}sinudu$

An important point to note is that the limits of integration are limits on the variable $t$ , not $u$ . To emphasise this they have been written explicitly as $t=2$ and $t=3$ . When we integrate with respect to the variable $u$ , the limits must be written in terms of $u$ . From the substitution $u=t^2$ , note that when $t=2$ then $u=4$ and when $t=3$ then $u=9$ so the integral becomes

$\frac{1}{2}{\int \; <!--nolimits\; -->}_{u=4}^{u=9}sinudu=\frac{1}{2}{\left[-cosu\right]}_4^9=\frac{1}{2}\left(-cos9+cos4\right)=0.129\text{to 3 d.p.}$

Exercise

Use suitable substitutions to find

1. ${\int \; <!--nolimits\; -->}_1^2{\left(2x+3\right)}^{7}dx$ ,
2. ${\int \; <!--nolimits\; -->}_0^{1}3t^2{e}^{t^3}dt$ .

Answer

1. $u=2x+3$ is suitable;   $3.359\times 10^5$  to 4 sig. figs.
2. 1.718  to 3 d.p.