Question

Solve: $\int \frac{x^2}{x^2+5}dx$

Answer 1

$\int \frac{x^{2}dx}{x^2+5}$

$=\int \frac{x^2+5-5}{x^2+5}dx$

$=\int (\frac{x^2+5}{x^2+5}-\frac{5}{x^2+5})dx$

$=\int (1-\frac{5}{x^2+5})dx$

$=\int (1-\frac{5}{x^2+(\sqrt{5}{)}^2})dx$

$=x-\frac{5}{\sqrt{5}}{\tan }^{-1}\left(\frac{x}{\sqrt{5}}\right)+c[\int \frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\frac{x}{a}]$

$=x-\sqrt{5}{\tan }^{-1}\left(\frac{x}{\sqrt{5}}\right)+c$

$\therefore \int \frac{x^2}{x^2+5}dx=x-\sqrt{5}{\tan }^{-1}\left(\frac{x}{\sqrt{5}}\right)+c$