Continuing from Hint 1, we get 2 x 2 + 3 x 2 + 1 = 2 ( x 2 + 1 ) + 1 x 2 + 1 = 2 + 1 x 2 + 1 {\displaystyle {\frac {2x^{2}+3}{x^{2}+1}}={\frac {2(x^{2}+1)+1}{x^{2}+1}}=2+{\frac {1}{x^{2}+1}}} . Now we have two terms, each of which has an antiderivative that we know.
