2 Definite integration

When dealing with definite integrals the relevant formula is as follows:

Example 17

Find ${\int \; <!--nolimits\; -->}_0^{2}x{\text{e}}^{x}dx$ .

Solution

We let $f=x$ and $g={\text{e}}^x$ . Then $\frac{df}{dx}=1$ and $\int \; <!--nolimits\; -->gdx={\text{e}}^x$ . Using integration by parts we obtain

${\int \; <!--nolimits\; -->}_0^{2}x{\text{e}}^{x}dx={\left[x{\text{e}}^x\right]}_0^2-{\int \; <!--nolimits\; -->}_0^{2}1.{\text{e}}^{x}dx=2{\text{e}}^2-{\left[{\text{e}}^x\right]}_0^2=2{\text{e}}^2-\left[{\text{e}}^2-1\right]={\text{e}}^2+1\text{(or 8.389 to 3 d.p.)}$

Sometimes it is necessary to apply the formula more than once as the next Example shows.

Example 18

Find the definite integral of $x^2{\text{e}}^x$ from 0 to 2; that is, find ${\int \; <!--nolimits\; -->}_0^2{x}^2{\text{e}}^{x}dx$ .

Solution

We let $f=x^2$ and $g={\text{e}}^x$ . Then $\frac{df}{dx}=2x\text{ and }\int \; <!--nolimits\; -->gdx={\text{e}}^x$ .  Using integration by parts:

${\int \; <!--nolimits\; -->}_0^2{x}^2{\text{e}}^{x}dx={\left[x^2{\text{e}}^x\right]}_0^2-{\int \; <!--nolimits\; -->}_0^{2}2x{\text{e}}^{x}dx=4{\text{e}}^2-2{\int \; <!--nolimits\; -->}_0^{2}x{\text{e}}^{x}dx$

The remaining integral must be integrated by parts also but we have just done this in the example above. So ${\int \; <!--nolimits\; -->}_0^2{x}^2{\text{e}}^{x}dx=4{\text{e}}^2-2\left[{\text{e}}^2+1\right]=2{\text{e}}^2-2=12.778\text{ (3 d.p.)}$

Task!

Find ${\int \; <!--nolimits\; -->}_0^{\pi ∕4}\left(4-3x\right)sinxdx$ .

What are your choices for $f,g$ ?

Answer

Answer

Exercises

1. Evaluate the following:
   1. ${\int \; <!--nolimits\; -->}_0^{1}xcos2xdx$ ,
   2. ${\int \; <!--nolimits\; -->}_0^{\pi ∕2}xsin2xdx$ ,
   3. ${\int \; <!--nolimits\; -->}_{-1}^{1}t{\text{e}}^{2t}dt$
2. Find ${\int \; <!--nolimits\; -->}_1^2\left(x+2\right)sinxdx$
3. Find ${\int \; <!--nolimits\; -->}_0^1\left(x^2-3x+1\right){\text{e}}^{x}dx$

Answer