Problem 27 - Essential Calculus

Question

Find or evaluate the integral. (Complete the square, if necessary.) $$ \int \frac{x+2}{\sqrt{-x^{2}-4 x}} d x $$

Step-by-Step Solution

Answer

The result to this integral is \(-\sqrt{(2-x)^{2}}+C\).

Step 1: Completing the Square

Firstly, the quadratic expression under the square root needs to be rearranged into complete square format. Notice that \(x^{2}+4x\) can be written as \((x+2)^{2}-4\).

Step 2: Rewrite the Exercise

The expression can now be rewritten as: \(\int \frac{x+2}{\sqrt{-(x+2)^{2}+4}} dx\).

Step 3: Variable Substitution

Now, use u-substitution to simplify the integral further. Let \(u=x+2\), so differentiating gives \(du=dx\). The integral can now be rewritten as:\(\int \frac{u}{\sqrt{-u^{2} +4}} du\).

Step 4: Simplify the Integral

Recognize this as a standard form of integral and simplify it further, which yields: \(-\sqrt{4-u^{2}}+C\), where C represents the constant of integration.

Step 5: Substitute back the original variable

Substitute \(u=x+2\) back into the equation to get the answer in terms of x:\(-\sqrt{(2-x)^{2}}+C\)