Problem 21 - Calculus: Graphical, Numerical, Algebraic

Question

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

Step-by-Step Solution

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 1: 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: \(\int \frac{3 du}{(u^{2}+3)^2}\).

Step 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 \(\sqrt{3} \cdot arctan(\frac{u}{\sqrt{3}})+C\).

Step 3: 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 \(\sqrt{3} \cdot arctan(\frac{x}{3}\sqrt{3})+C\)

Step 4: 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.

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify the process of integration. This method involves changing variables in an integral to make it easier to solve.
This is especially useful when dealing with complicated integrands.In the current exercise, we use the substitution \(u = \frac{x}{3}\). This requires expressing \(x\) and \(dx\) in terms of \(u\).

• First, express \(x\) as \(x = 3u\).
• Then, differentiate this relation to find \(dx = 3du\).

We now rewrite the entire integral in terms of \(u\):

• Replace \(x\) to get: \(\int \frac{3du}{(u^2+3)^2}\).

This transformation simplifies the integration process.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \(arctan\), appear often in integration problems, especially when dealing with trigonometric substitutions.
These functions are essential for finding antiderivatives involving variable changes or specific integrands.In this exercise, after performing the substitution method, the integral simplifies to one that resembles the derivative of the arctan function.

• The integral \(\int \frac{1}{u^2+3} \, du\) leads to the antiderivative \(arctan\left(\frac{u}{\sqrt{3}}\right)\).

The presence of the square root and constant factors shows how inverse trigonometric functions can cleverly simplify integration solutions. After back-substituting \(u\), we express the final result using these inverse trigonometric functions.

Verification by Differentiation

Verification by differentiation is a critical step in confirming the accuracy of an integral solution. It involves taking the derivative of the resulting function to check if you get back the original integrand.
If they match, it indicates the solution is correct.In our context, after finding the antiderivative:

• We differentiate \(\sqrt{3} \cdot arctan\left(\frac{x}{3\sqrt{3}}\right) + C\).
• The derivative should bring us back to \(\frac{1}{x^2+9}\).

Seeing that the differentiation of our solution matches the original integrand reassures us of the solution’s correctness. This step closes the loop on the problem by tying together integration and differentiation concepts.