Final answer to the problem
Step-by-step Solution
Specify the solving method
Find the integral
We can solve the integral $\int\frac{\left(1-x^2\right)^2}{x^2+2x+1}dx$ by applying integration method of trigonometric substitution using the substitution
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
Substituting in the original integral, we get
Applying the trigonometric identity: $1-\sin\left(\theta \right)^2 = \cos\left(\theta \right)^2$
Simplify $\left(\cos\left(\theta \right)^2\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $2$
When multiplying exponents with same base you can add the exponents: $\cos\left(\theta \right)^{4}\cos\left(\theta \right)$
The trinomial $\sin\left(\theta \right)^2+2\sin\left(\theta \right)+1$ is a perfect square trinomial, because it's discriminant is equal to zero
Using the perfect square trinomial formula
Factoring the perfect square trinomial
Rewrite the trigonometric function $\cos\left(\theta \right)^{5}$ as the product of two lower exponents
Rewrite $\cos\left(\theta \right)^{4}$ as by applying trig identities
We can solve the integral $\int\frac{\left(1-\sin\left(\theta \right)^2\right)^{2}\cos\left(\theta \right)}{\left(\sin\left(\theta \right)+1\right)^{2}}d\theta$ 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 $\sin\left(\theta \right)$ 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 $d\theta$ 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 $d\theta$ in the previous equation
Substituting $u$ and $d\theta$ in the integral and simplify
Factor the difference of squares $\left(1-u^2\right)$ as the product of two conjugated binomials
We can solve the integral $\int\left(1-u\right)^{2}du$ 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 $v$), which when substituted makes the integral easier. We see that $1-u$ it's a good candidate for substitution. Let's define a variable $v$ and assign it to the choosen part
Now, in order to rewrite $du$ in terms of $dv$, we need to find the derivative of $v$. We need to calculate $dv$, we can do that by deriving the equation above
Isolate $du$ in the previous equation
Substituting $v$ and $du$ 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 $v$ with the value that we assigned to it in the beginning: $1-u$
Replace $u$ with the value that we assigned to it in the beginning: $\sin\left(\theta \right)$
Express the variable $\theta$ in terms of the original variable $x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$