Problem 29 - Calculus: One and Several Variables

Question

Calculate. $$\int \frac{d x}{\left(x^{2}-4 x+4\right)^{3 / 2}}$$.

Step-by-Step Solution

Answer

The solution to the given integral is: \(\frac{1}{2}(x - 2)^{-2} + C\).

Step 1: Rewrite the quadratic expression as a perfect square

First, we need to rewrite the denominator of the given integral as a perfect square. Let's factor the quadratic expression \(x^2 - 4x + 4\): \(x^2 - 4x + 4 = (x - 2)^2\) Now, the integral becomes: \[ \int \frac{dx}{((x - 2)^2)^{3/2}}. \]

Step 2: Perform a substitution

Let's use the substitution method with \(u = x-2\). Then, we have \(du = dx\). When we rewrite the integral using this substitution, we will get: \[ \int \frac{du}{(u^2)^{3/2}}. \]

Step 3: Integrate the integral

Now, we can easily integrate the resulting integral: \[ \int \frac{du}{u^3} = -\frac{1}{2}\int u^{-3} du. \] Now we find the antiderivative of the integrand: \[ -\frac{1}{2}\int u^{-3} du = -\frac{1}{2} \frac{u^{-2}}{-2} + C = \frac{1}{2}u^{-2} + C. \] Now we can replace \(u\) with the original variable \(x - 2\): \[ \frac{1}{2}(x - 2)^{-2} + C. \] Hence, the solution to the given integral is: \[ \int\frac{dx}{(x^2 - 4x + 4)^{3 / 2}} = \frac{1}{2}(x - 2)^{-2} + C. \]

Key Concepts

Integration by Substitution

Integration by substitution, often regarded as the reverse process of the chain rule in differentiation, is a method to simplify complex integrals. When facing an integral that does not have a straightforward antiderivative, substituting part of the integral with a new variable (often denoted as 'u') can unravel a simpler form that is easier to integrate.

For example, when the integrand is a composite function like a polynomial raised to a power, we can substitute the polynomial itself with 'u', which after differentiating gives us 'du'. This turns the integral into a form in terms of 'u', which we can integrate more easily.

Applying Substitution

In our exercise, we used the substitution \( u = x-2 \) for the integrand's denominator. The differential \( du = dx \) ensures the substitution aligns perfectly, converting our integrand into a much simpler \( u \) term expression. After the substitution, the process of finding the antiderivative becomes much more manageable.

Antiderivative

An antiderivative is a function whose derivative yields the original function. In simpler terms, it's the reverse process of differentiation. If you differentiate an antiderivative, you should get back the function you started with. In the realm of integrals, particularly indefinite integrals, antiderivatives play a crucial role as they are essentially the function you get after integrating an expression.

Application to the Exercise

By finding the antiderivative, we reversed the derivative process, arriving at the expression \( -\frac{1}{2}u^{-2} \) for our integral after applying substitution. By including the integration constant, 'C', we acknowledge that there are infinitely many antiderivatives for a given function, each differing by just a constant value.

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression. It takes the form \( (ax + b)^2 \) and expands into an expression like \( a^2x^2 + 2abx + b^2 \). Recognizing and factoring perfect square trinomials allows us to greatly simplify integration problems by reducing complex polynomials to forms that are easier to work with.

Perfect Square in Our Example

The original integral contained the quadratic expression \( x^2 - 4x + 4 \), which is a perfect square trinomial and can be written as \( (x-2)^2 \). This simplification was the initial step that enabled us to apply the substitution method effectively, as it transformed the expression into a form that was recognizable and directly applicable for the substitution method.