Problem 34 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int \frac{x^{2}}{(x-2)^{3}} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral \(\int \frac{x^2}{(x-2)^3} dx\). Answer: \(\ln{|x-2|} - \frac{4}{x-2} - 2(x-2)^{-2} + C\)

Step 1: Choose substitution variable

Let \(u = (x - 2)\). We will substitute this variable into the integral to simplify it.

Step 2: Find the differential of u

Now we need to find \(du\), which is the derivative of \(u\) with respect to \(x\). $$du = \frac{d}{d x}(x - 2) dx = dx$$

Step 3: Rewrite the integral using the substitution

We need to express \(x^2\) and \(dx\) in terms of \(u\). From our substitution variable, we have \(x = u + 2\). Therefore, $$x^2 = (u+2)^2 = u^2 + 4u + 4$$ Next, substitute these values into the integral and rewrite it in terms of \(u\): $$\int \frac{x^{2}}{(x-2)^{3}} d x=\int \frac{u^{2}+4 u+4}{u^{3}} d u$$

Step 4: Split the integral into simpler terms

To make it easier to integrate, we will separate the fraction into three separate terms: $$\int \frac{u^{2}+4 u+4}{u^{3}} d u = \int \frac{u^2}{u^3} du + 4\int \frac{u}{u^3} du + 4\int \frac{1}{u^3} du$$

Step 5: Simplify the terms and integrate

Now we have three simpler integrals that we can integrate easily: $$\int \frac{u^2}{u^3} du + 4\int \frac{u}{u^3} du + 4\int \frac{1}{u^3} du = \int u^{-1} du + 4\int u^{-2} du + 4\int u^{-3} du$$ Integrating these terms, we get: $$ \ln{|u|} + 4\left(-\frac{1}{u}\right) - 2u^{-2} + C, $$ where C is the constant of integration.

Step 6: Substitute x back and simplify

Now, substitute \(x\) back in place of \(u\), using the substitution variable (\(u = x-2\)): $$ \ln{|x-2|} - \frac{4}{x-2} - 2(x-2)^{-2} + C $$ This is the final result: $$ \boxed{\ln{|x-2|} - \frac{4}{x-2} - 2(x-2)^{-2} + C}. $$

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus used to simplify complex integrals. It works by introducing a new variable to replace a part of the integral, making it easier to solve. When using the substitution method, you begin by identifying a part of the integrand that will simplify the expression when replaced. In this exercise, we chose \( u = (x-2) \). This substitution transforms the original variable, \( x \), into \( u \), simplifying the integration process by breaking it into smaller, more manageable parts.
Once you've chosen your substitution, the next steps are:

• Find the derivative of your new variable in terms of the original variable (\( du = dx \)).
• Express all parts of the integral in terms of the new variable \( u \).
• Solve the transformed integral.

Choosing the correct substitution can greatly ease the process and lead to a more straightforward integration.

Indefinite Integral

An indefinite integral represents a family of functions whose derivative is the integrand. Unlike definite integrals, which result in a number, indefinite integrals yield a function plus a constant, \( C \).The notation \( \int f(x) \, dx \) is used to signify the indefinite integral of \( f(x) \). This process is essentially the opposite of differentiation.
In our exercise, after applying the substitution, we converted the indefinite integral \( \int \frac{x^{2}}{(x-2)^{3}} \, dx \) into simpler terms. This entire expression is considered an indefinite integral until a specific point or boundary values are applied.It's crucial to always add the constant of integration, \( C \), when calculating indefinite integrals. This constant accounts for any potential vertical shifts in the antiderivative function since integration can reveal multiple possible solutions corresponding to any constant added.

Integration Techniques

Integration techniques are various methods used to find integrals of functions. The substitution method is one such technique, but there are others, such as integration by parts and partial fraction decomposition.In this exercise, we demonstrated splitting the fraction into simpler terms to make the integration process more straightforward:\[\int \frac{u^{2}+4 u+4}{u^{3}} \, du = \int u^{-1} \, du + 4\int u^{-2} \, du + 4\int u^{-3} \, du\]Each term is now easier to integrate compared to the original complex integral. The goal of using various integration techniques is to translate a complex integral into one or more simpler integrals that can be computed easily. Practicing these techniques enhances your ability to solve intricate problems with precision.