Question

In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int \frac{d x}{x^{2}+9}, \quad u=\frac{x}{3}$$

Short answer

The solution to the integral \(\int \frac{1 dx}{x^{2}+9}\) when \(u=\frac{x}{3}\) is given by \(\sqrt{3} \cdot arctan(\frac{x}{3}\sqrt{3})+C\). Verification by derivative confirms the solution.

Step-by-step solution

Perform the Replacement

Let's start by substituting \(u = x/3\). This substitution results in \(x = 3u\). We must change the \(dx\) along with \(x\) to transform the integral completely in terms of \(u\). So, \(dx = 3du\). Now we have the integral in the form: \(\int \frac{3 du}{(u^{2}+3)^2}\).

Integral Solving

The integral becomes a simple rational function: \(\int \frac{3 du}{(u^{2}+3)^2}\). The integrand is now an even power of a simple sum and simplifies to \(arctan\). The integral becomes: \(3 \int \frac{1 du}{u^2+3}\) which is equal to \(arctan(u/\sqrt{3})\) plus a constant. It simplifies as \(\sqrt{3} \cdot arctan(\frac{u}{\sqrt{3}})+C\).

Back substitution

Replace \(u\) by \(x/3\) back into the equation, it becomes: \(\sqrt{3} \cdot arctan(\frac{x}{3\sqrt{3}})+C\) or normalize \(\sqrt{3} \cdot arctan(\frac{x}{3}\sqrt{3})+C\)

Verification by Derivative

To verify the solution, the derivative of \(\sqrt{3} \cdot arctan(\frac{x}{3\sqrt{3}})+C\) is computed. The derivative of \(\sqrt{3} \cdot arctan(\frac{x}{3\sqrt{3}})\) is \(\frac{1}{x^{2}+9}\), which confirms the solution.