Problem 67 - Thomas Calculus in SI Units

Question

Right, or wrong? Give a brief reason why. $$\int \frac{-15(x+3)^{2}}{(x-2)^{4}} d x=\left(\frac{x+3}{x-2}\right)^{3}+C$$

Step-by-Step Solution

Answer

The statement is right because differentiation of the RHS equals the integrand.

Step 1: Examine the Given Integral

We need to evaluate the integral \( \int \frac{-15(x+3)^{2}}{(x-2)^{4}} \, dx \). The expression suggests using a substitution method, especially considering the structure of the integral. A possible substitution could be \( u = \frac{x+3}{x-2} \).

Step 2: Differentiate and Apply Substitution

Let's try differentiating \( u = \frac{x+3}{x-2} \), which gives \( du = \left( \frac{(x-2) - (x+3)}{(x-2)^2} \right) dx = \frac{-5}{(x-2)^2} dx \). However, this doesn't directly match the original integrand, and the transformation seems overly complex without simplifying our task. Instead, let's check the differentiation of the right-hand side of the equation.

Step 3: Differentiate the Right-Hand Side

We differentiate the expression \( \left(\frac{x+3}{x-2}\right)^{3} \) with respect to \( x \). By using the chain rule, \( \frac{d}{dx} \left(\frac{x+3}{x-2}\right)^{3} = 3\left(\frac{x+3}{x-2}\right)^{2}\cdot \frac{d}{dx}\left(\frac{x+3}{x-2}\right) \). Calculating \( \frac{d}{dx}\left(\frac{x+3}{x-2}\right) \) gives \( \frac{-5}{(x-2)^{2}} \), leading to the derivative \( \frac{-15(x+3)^{2}}{(x-2)^{4}} \).

Step 4: Verify the Result

The derivative \( \frac{-15(x+3)^{2}}{(x-2)^{4}} \) matches the original integrand. Therefore, integrating \( \frac{-15(x+3)^{2}}{(x-2)^{4}} \) gives \( \left(\frac{x+3}{x-2}\right)^{3} + C \), verifying the original statement is indeed correct.

Step 5: Conclusion

The given integral statement is correct, as both the method of substitution and direct differentiation confirms the right-hand side expression matches the antiderivative of the integrand.

Key Concepts

Integration Techniques

In integral calculus, it's essential to identify the correct technique to simplify and solve integrals. This often involves recognizing a pattern or structure in the integrand that matches known formulas or lends itself to modification. The given integral \[\int \frac{-15(x+3)^{2}}{(x-2)^{4}} \, dx\]might first suggest using techniques like substitution, comparison with standard forms, or partial fraction decomposition if applicable.

• Substitution Method: This technique replaces a complicated part of the integrand with a single variable, simplifying the integration process.
• Direct Differentiation Verification: An integration technique that involves checking potential antiderivatives by differentiating them to see if they match the original integrand.

Both substitution and direct differentiation are useful in verifying the solution to integrals. They help confirm that the process and result are valid and accurate.

Chain Rule

The chain rule is a fundamental concept in calculus, used primarily for differentiating compositions of functions. It's crucial in understanding how to handle integrals with nested functions.For example, in the process of verifying our original integral statement, the chain rule helps us take the derivative of the proposed solution on the right-hand side:\[\frac{d}{dx} \left( \frac{x+3}{x-2} \right)^{3}\]To apply the chain rule:

• First, differentiate the outer function with respect to its inner function: \[3\left(\frac{x+3}{x-2}\right)^2\]
• Next, multiply this by the derivative of the inner function as applied to \(x\).

This step requires calculating \[\frac{d}{dx}\left(\frac{x+3}{x-2}\right)\]which due to the quotient rule, gives the derivative necessary for further simplification and validation.

Substitution Method

The substitution method is like renaming a complex part of the function to make it easier to integrate. In the context of our problem, the substitution method was initially considered as a way to simplify the integral.A potential substitution might have been: \[u = \frac{x+3}{x-2}\]This substitution would be expected to simplify the integrand into something more easily integrable. However, after trying to change variables, the resultant new integral didn't simplify directly or provide a straightforward path to solution compared to verifying through differentiation.Nevertheless, substitution remains a powerful integration technique whenever the integrand contains composite functions that can be neatly untangled with a substitution.

Antiderivative Verification

Antiderivative verification serves as a neat way of confirming your work when solving integrals. It involves differentiating your supposed answer to see if it matches the original integrand.When we presumed the antiderivative of the original integral was:\[\left(\frac{x+3}{x-2}\right)^{3} + C\]To verify, we differentiated. The resulting expression:\[\frac{-15(x+3)^{2}}{(x-2)^{4}}\]matched our original integrand perfectly. This confirms that our antiderivative is correct. Verification using differentiation not only strengthens confidence in your solution but also provides a robust method to cross-check complex results. In scenarios where substitution doesn't easily resolve the integrand, this technique shines by affirming the correctness of solutions.