Final Answer
Step-by-step Solution
Specify the solving method
We can solve the integral $\int\sqrt{2w+1}dw$ 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 $2w+1$ 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 $dw$ 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 $dw$ in the previous equation
Substituting $u$ and $dw$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
Divide $1$ by $2$
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 $\frac{1}{2}$
Simplify the expression inside the integral
Replace $u$ with the value that we assigned to it in the beginning: $2w+1$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$