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

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

Transcript text: Simplify. \[ \int \frac{x^{3}}{x^{2}+4 x+4} d x \]

Solution

Step 1: Factor the Denominator

The given integral is

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

First, we factor the quadratic denominator:

\[ x^{2} + 4x + 4 = (x + 2)^{2}. \]

Step 2: Rewrite the Integral

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

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

Step 3: Perform the Integration

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.

Thus, the final result of the integral is

\[ \boxed{\frac{x^{2}}{2} - 4x + 12 \log(x + 2) + \frac{8}{x + 2} + C}. \]

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