Simplify $\left(\sqrt{1+x}\right)^4$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $\frac{1}{2}$ and $n$ equals $4$
Expand
Divide $x^3$ by $1+2x+x^2$
Resulting polynomial
Simplify the expression
We can solve the integral $\int\frac{3}{1+x}dx$ 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 $1+x$ 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 finding the derivative of the equation above
Substituting $u$ and $dx$ in the integral and simplify
The integral $\int xdx$ results in: $\frac{1}{2}x^2$
The integral $\int-2dx$ results in: $-2x$
The integral $\int\frac{3}{u}du$ results in: $3\ln\left(1+x\right)$
The integral $\int\frac{-1}{\left(1+x\right)^{2}}dx$ results in: $\frac{1}{x+1}$
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 $C$
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