Solutions - 0060-04

(1) Perform long division:

$$\frac{x^2}{x^2+9}\gg \gg 1-\frac{9}{x^2+9}$$

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(2) Use $\arctan$ to integrate:

Recall formula:

$$\int \frac{dx}{x^2+h^2}=\frac{1}{h}{\tan }^{-1}\left(\frac{x}{h}\right)+C$$

Choose $h=3$. Then:

$$\int \frac{9}{x^2+9}dx\gg \gg 9\frac{1}{3}{\tan }^{-1}\left(\frac{x}{3}\right)+C$$

The final answer is therefore:

$$x-3{\tan }^{-1}\left(\frac{x}{3}\right)+C$$