Problem 31 - Essential Calculus

Question

Complete the square and find the integral. $$ \int \frac{1}{\sqrt{4 x-x^{2}}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(-ln |2 - x| + C\).

Step 1: Rewrite the square root

The first goal is to complete the square for \(4x - x^2\). This can be written as -\((x-2)^2\). As a result, the integral becomes: \(\int \frac{1}{\sqrt{-(x-2)^2}} dx\). However, we can simplify this by taking the negative sign out of the square root, so we have: \(\int \frac{1}{\sqrt{(2-x)^2}} dx\)

Step 2: Substitute variables

Now, a substitution is done: Let \(u = 2-x\). Then \(du = -dx\). We replace \(dx\) with \(-du\) in the integral, and we have: \(\int \frac{-1}{\sqrt{u^2}} du\). This is the same as: \(\int \frac{-du}{u}\).

Step 3: Apply integral formula

The integral above can now be solved by applying the formula for the natural log integral: \(\int \frac{du}{u} = ln |u| + C\). Thus, \(\int \frac{-du}{u} = -ln |u| + C\). Substitute \(u = 2 - x\) back into the equation: \(-ln |2 - x| + C\)