We can solve the integral $\int2\sqrt{-\left(x+\frac{5}{2}\right)^2+4}dx$ by applying integration method of trigonometric substitution using the substitution

The power of a product is equal to the product of it's factors raised to the same power

Factoring by $4$

The power of a product is equal to the product of it's factors raised to the same power

Applying the trigonometric identity: $1-\sin\left(\theta \right)^2 = \cos\left(\theta \right)^2$

The integral of a function times a constant ($4$) is equal to the constant times the integral of the function

Simplify $\sqrt{\cos\left(\theta \right)^2}$ 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 $\frac{1}{2}$

When multiplying two powers that have the same base ($\cos\left(\theta \right)$), you can add the exponents

Apply the formula: $\int\cos\left(\theta \right)^2dx$$=\frac{1}{2}\theta +\frac{1}{4}\sin\left(2\theta \right)+C$, where $x=\theta $

Express the variable $\theta$ in terms of the original variable $x$

Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$

Multiply the fraction and term in $2\left(\frac{1}{4}\right)\sin\left(\theta \right)\cos\left(\theta \right)$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$