Problem 38 - Calculus of a Single Variable

Question

Completing the Square In Exercises \(33-42,\) find or evaluate the integral by completing the square. $$ \int \frac{2}{\sqrt{-x^{2}+4 x}} d x $$

Step-by-Step Solution

Answer

-2arcsin((x-2)/2) + C

Step 1: Complete the Square

In order to integrate the function under the integral sign, we first need to complete the square for the quadratic expression under the square root. The original function under the integral can be rewritten as follows: \( \int \frac{2}{\sqrt{-(x-2)^{2}}}\, dx \). After completing the square, the coefficient -1 outside the bracket is coupled with the integral to give the final expression: \( -\int \frac{2}{\sqrt{(x-2)^{2}}}\, dx \)

Step 2: Integral Evaluation

Now, the integral of this function is a standard result known from the table of integrals. The integral of \( \frac{1}{\sqrt{a^2-x^2}} \) is \( \arcsin{\frac{x}{a}} \). Applied to our problem, it becomes: \( -2\arcsin{\frac{x-2}{2}} \)

Step 3: Add Constant of Integration

Finally, in an indefinite integral, there's always a constant of integration to account for the 'initial condition' since the antiderivative is only determined up to a constant. Hence the final result will look like: \( -2\arcsin{\frac{x-2}{2}} + C \)

Key Concepts

Indefinite Integral

An indefinite integral, also known as an antiderivative, represents a family of functions that reverses the process of differentiation. It's the act of finding a function whose derivative is the given function. In the given exercise, the integral sign \( \int \) instructs us to find all functions whose derivative would result in the integrand—here, the function \( \frac{2}{\sqrt{-x^{2}+4x}} \).

When solving an indefinite integral, the result is not a single function but a set of functions that differ by a constant, as any constant becomes zero when differentiated. The notation for an indefinite integral includes a '+ C' at the end, where 'C' represents the constant of integration. This constant is essential since multiple functions can have the same derivative, and it ensures the general solution of the integral includes all possibilities.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically written in the form \( ax^2+bx+c \). In the exercise, the quadratic expression inside the square root \(\sqrt{-x^{2}+4x}\) posed a challenge for integration. To address this, completing the square is a technique used to transform a quadratic expression into a perfect square trinomial \( (x-h)^{2} + k \), where \( h \) and \( k \) are constants. This makes the expression easier to integrate.

Completing the square involves finding a constant that, when added and subtracted to the expression, forms a perfect square plus some remainder. In this case, we completed the square to transform \( -x^{2}+4x \) into a perfect square \( -(x-2)^{2} \), making the process of finding the indefinite integral more straightforward. It is a powerful tool for integrating a class of functions that would otherwise be difficult to handle.

Antiderivative

The term antiderivative refers to the inverse operation of differentiation. Finding an antiderivative means identifying a function \( F(x) \) such that \( F'(x) = f(x) \) for a given function \( f(x) \). When calculating the indefinite integral of a function, what we’re really doing is finding its antiderivatives.

For the quadratic expressions present in this exercise, specifically within square roots, recognizing patterns that relate to standard antiderivatives is necessary. The antiderivative of \( \frac{1}{\sqrt{a^2-x^2}} \) is a known pattern, with the result being \( \arcsin{\frac{x}{a}} \). This recognition is often facilitated by standard integral tables or prior knowledge of antiderivatives, enabling us to solve integrals that at first may seem complex. In the context of the exercise, this knowledge leads to the final antiderivative result, expressed as part of a family of functions, distinguished by the '+ C' indicative of the constant of integration.