Problem 6 - Essential Calculus

Question

Find the indefinite integral. $$ \int \frac{x}{\sqrt{9-x^{2}}} d x $$

Step-by-Step Solution

Answer

\(-\sqrt{9 - x^2} + C\)

Step 1 : Set up the substitution

Identify the inner function which is more complicated and could simplify the integral when set as \(u\). The inner function here is \(9-x^2\). Therefore, let \(u = 9 - x^2\). Now, compute the derivative, \(du = -2xdx\). From this, solve for \(dx\), which will be used to substitute \(dx\) in the integral. Therefore, \(dx = - \frac{1}{2x} du\).

Step 2 : Substitute in the integral

Replace \(9 - x^2\) with \(u\) and \(dx\) with \(-\frac{1}{2x} du\) in the integral. This results in \(-\frac{1}{2} \int \frac{1}{\sqrt{u}} du\).

Step 3 : Evaluate the integral

Now, evaluate the integral. The integral of \(\frac{1}{\sqrt{u}}\) is \(2\sqrt{u}\). Therefore, the integral becomes -\(\sqrt{u}\).

Step 4 : Reverse the substitution

Substitute back \(u\) to bring the integral back in terms of \(x\). As we started with \(u = 9- x^2\), replacing gives \(-\sqrt{9 - x^2}\).

Step 5 : Add constant of integration

One thing to remember about indefinite integrals is to include the constant of integration, \(C\), at the end of the solution. This gives the final answer as \(-\sqrt{9 - x^2} + C\).