Final Answer
Step-by-step Solution
Specify the solving method
We can solve the integral $\int\left(x+2\right)^4\ln\left(x+2\right)dx$ 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
We can solve the integral $\int\left(x+2\right)^4dx$ 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 $x+2$ 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
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 $4$
Replace $u$ with the value that we assigned to it in the beginning: $x+2$
Now replace the values of $u$, $du$ and $v$ in the last formula
The integral $-\int\frac{\left(x+2\right)^{4}}{5}dx$ results in: $-\frac{1}{25}\left(x+2\right)^{5}$
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 $C$