Problem 18 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int \frac{1}{(3 x+1)^{2}} d x$$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{3(3x+1)} + C\); differentiation confirms this result.

Step 1: Identify the integral form

We need to find the integral of \( \int \frac{1}{(3x+1)^2} \, dx \). This is a standard form that can be solved using substitution.

Step 2: Use substitution

Let \( u = 3x + 1 \). Then, the derivative \( du = 3 \, dx \), or \( dx = \frac{1}{3} \, du \). Substitute these into the integral.

Step 3: Substitute and simplify the integral

Substituting \( u \) and \( dx \) in the integral, we have \( \int \frac{1}{u^2} \frac{1}{3} \, du = \frac{1}{3} \int u^{-2} \, du \).

Step 4: Integrate using power rule

The integral \( \int u^{-2} \, du \) is \( -u^{-1} = -\frac{1}{u} \). Therefore, \( \frac{1}{3} \int u^{-2} \, du = -\frac{1}{3u} + C \).

Step 5: Substitute back the original variable

Substitute \( u = 3x + 1 \) back into the equation: \(-\frac{1}{3(3x+1)} + C \).

Step 6: Differentiate to check the solution

Differentiate \(-\frac{1}{3(3x+1)} + C \) with respect to \( x \). Using the chain rule, \( \frac{d}{dx} \left( -\frac{1}{3(3x+1)} \right) = \frac{1}{(3x+1)^2} \), which matches the original integrand.

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus for simplifying complex integrals. It involves changing variables to make the integral easier to evaluate.
In our example, the integral is \( \int \frac{1}{(3x+1)^2} \, dx \). This can be simplified by using substitution.

• First, we choose a substitution that will transform the integral into a simpler form. Let's set \( u = 3x+1 \), which simplifies the expression inside the integral.
• Next, compute the differential: \( du = 3 \, dx \) or equivalently, \( dx = \frac{1}{3} \, du \).
• By substituting these into the integral, we transform it into \( \int \frac{1}{u^2} \frac{1}{3} \, du = \frac{1}{3} \int u^{-2} \, du \).

This method converts the original variable \( x \) into a new variable \( u \), helping to solve the integral more easily. The key is to choose \( u \) such that the integral becomes one that is straightforward to integrate.

Integral Verification

Once we find the integral, it's vital to verify the solution by differentiation. This confirms that our answer is correct.
Here's how we do it in the given problem:

• After integrating, our solution was \( -\frac{1}{3(3x+1)} + C \).
• To verify, we differentiate this result with respect to \( x \).
• Applying basic differentiation rules, we use the chain rule to get \( \frac{d}{dx} \left( -\frac{1}{3(3x+1)} \right) \).
• This differentiation simplifies back to \( \frac{1}{(3x+1)^2} \), which is exactly the original function we integrated.

Verification is an essential step as it solidifies our solution and ensures it aligns with the given integrand. The process involves rediscovering the original function, proving the integration was performed correctly.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate compositions of functions. It is especially useful in verifying integrals, as demonstrated in this solution.
When differentiating \( -\frac{1}{3(3x+1)} + C \), the chain rule helps tackle the inner function \( 3x+1 \). For functions of the form \( f(g(x)) \), the chain rule states:

• Take the derivative of the outer function \( f \) with respect to its argument \( g(x) \).
• Multiply this by the derivative of the inner function \( g(x) \) with respect to \( x \).
• In the problem at hand: The outer function is \(-\frac{1}{u}\), and the inner function is \(3x+1\).

Differentiating gives us the derivative \( \frac{1}{(3x+1)^2} \), proper execution of the chain rule yields results consistent with our original integrand. Thankfully, learning the chain rule aids not just in differentiation but also in understanding the transformations needed for integral verification.