Final answer to the problem
Step-by-step Solution
Specify the solving method
Find the integral
Rewrite the expression $\frac{\left(1-x^2\right)^2}{x^2+2x+1}$ inside the integral in factored form
Factor the difference of squares $\left(1-x^2\right)$ as the product of two conjugated binomials
We can solve the integral $\int\left(1-x\right)^2dx$ 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 $1-x$ 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
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
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 $2$
Replace $u$ with the value that we assigned to it in the beginning: $1-x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$