Final answer to the problem
Step-by-step Solution
Specify the solving method
Find the integral
Rewrite the fraction $\frac{\left(1-x^2\right)^2}{x^2+2x+1}$ inside the integral as the product of two functions: $\left(1-x^2\right)^2\frac{1}{x^2+2x+1}$
We can solve the integral $\int\left(1-x^2\right)^2\frac{1}{x^2+2x+1}dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
The trinomial $x^2+2x+1$ is a perfect square trinomial, because it's discriminant is equal to zero
Using the perfect square trinomial formula
Factoring the perfect square trinomial
We can solve the integral $\int\frac{1}{\left(x+1\right)^{2}}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x+1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Substituting $u$ and $dx$ in the integral and simplify
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Replace $u$ with the value that we assigned to it in the beginning: $x+1$
Now replace the values of $u$, $du$ and $v$ in the last formula
The integral $4\int\frac{\left(1-x^2\right)x}{-x-1}dx$ results in: $-2x^2-\frac{4}{3}x^{3}$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$