Questions: Simplify. [ int fracx^3x^2+4 x+4 d x ]

Solution Steps

The given integral is

$$\int \frac{x^{3}}{x^{2}+4 x+4} d x.$$

First, we factor the quadratic denominator:

$$x^{2} + 4x + 4 = (x + 2)^{2}.$$

Now, we can rewrite the integral using the factored form of the denominator:

$$\int \frac{x^{3}}{(x + 2)^{2}} d x.$$

The result of the integration is:

$$\frac{x^{2}}{2} - 4x + 12 \log(x + 2) + \frac{8}{x + 2} + C,$$

where $C$ is the constant of integration.

Final Answer