Question

$\int \frac{x^4 + 4x^2 + 4x + 4}{x^2 + 2}.dx$

Answer 1

$\frac{x^3}{3} + 2x + 2 \ln(x^2 + 2) + C$

Answer 2

The integral of a rational function where the numerator's degree is greater than the denominator's degree is solved by first simplifying the integrand. We factor the numerator to $(x^2+2)^2 + 4x$, which allows us to split the fraction into $(x^2+2) + \frac{4x}{x^2+2}$. The first term integrates directly to $\frac{x^3}{3} + 2x$. The second term is solved by substitution, letting $u = x^2+2$, which transforms the integral into $2\int \frac{1}{u}du = 2\ln|u|$. Substituting back, we get $2\ln(x^2+2)$. Combining these results yields the final answer.