Question:

<p>$x^3 + x^2 + x + 1 = x(x^2 + 2x + 3) - 1(x^2 + 2x + 3) + 4$</p><p>Hence $\int \frac{x^3 + x^2 + x + 1}{\sqrt{x^2 + 2x + 3}} dx = \int (x - 1) \sqrt{x^2 + 2x + 3} dx + \int \frac{4}{\sqrt{x^2 + 2x + 3}} dx$</p><p>$= \frac{1}{2} \int (2x + 2) \sqrt{x^2 + 2x + 3} dx - 2 \int \sqrt{x^2 + 2x + 3} dx + \int \frac{4}{\sqrt{x^2 + 2x + 3}} dx$</p><p>$= \frac{1}{2} \times \frac{2}{3} (x^2 + 2x + 3)^{3/2} - (x + 1) \sqrt{x^2 + 2x + 3} + 2 \ln \left| x + 1 + \sqrt{x^2 + 2x + 3} \right| + C$</p>