<s>[INST] Evaluating $\int^1_0\frac{x^2}{4x+5}dx$ How can I integrate this function ?
$$\int^1_0\frac{x^2}{4x+5}dx$$
 [/INST] Divide the $x^2$ by $4x+5$, using ordinary division of polynomials. We get
$$\frac{x^2}{4x+5}=\frac{1}{4}x-\frac{5}{16}+\frac{25}{16}\frac{1}{4x+5}.$$
Now the integration should be straightforward.
Alternately, as a second choice, let $u=4x+5$.  Then $du=4\,dx$, so $dx=\frac{1}{4}du$. Also, $x=\frac{1}{4}(u-5)$, so $x^2=\frac{1}{16}(u^2-10u+25)$.  We end up with
$$\int_{u=5}^9 \frac{1}{64}\frac{u^2-10u+25}{u}\,du.$$
 The integration is easy, because of the cancellations.
</s>