Set $y\=2 x\+1$ so that $\mathrm{dy}\=2 \mathrm{dx}$ and $\mathrm{dx}\=\mathrm{dy}\/2$. Under this change of variable, the given indefinite integral becomes

 $\int y^8 \frac{\mathrm{dy}}{2}\=\frac{1}{2}\frac{y^9}{9}\={\left(2 x\+1\right)}^9\/18$ 

Of course, there are settings in which the addition of an arbitrary constant is deemed essential.

Although it would be possible to expand the numerator and integrate termwise, it is not a recommended approach to integration! In fact, termwise integration produces a result that does not factor to the compact expression obtained above because of a missing additive constant.

In fact, termwise integration produces a result that does not factor to the compact expression obtained above because of a missing additive constant of $1\/18$. This can be seen by expanding ${\left(2 x\+1\right)}^9\/18$, that is, from

${\left(2 x\+1\right)}^9\/18$$\stackrel{\text{expand}}{\=}$$\frac{256}{9}{x}^9\+128x^8\+256x^7\+\frac{896}{3}{x}^6\+224x^5\+112x^4\+\frac{112}{3}{x}^3\+8x^2\+x\+\frac{1}{18}$

(which was done in Maple by applying Expand from the Context Panel.)

The final step, Revert, isn't taken in Figure 4.4.2(a) because Maple expands ${\left(2 x\+1\right)}^9\/18$, stretching the whole display and thereby putting the annotations "off the screen."$$