Problem 22 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int \frac{3}{4 x^{2}+4} d x$$

Step-by-Step Solution

Answer

The general antiderivative of the given function is \(\frac{3}{4}\arctan(x) + C\).

Step 1: Identify Type Of Function

The given function is a rational fraction. Note that the denominator is a perfect square. Remember that the derivative of \(\arctan(x)\) is \(\frac{1}{1+x^2}\). The function we have is a constant multiple of that derivative.

Step 2: Write Down Integral

Given integral is \(\int \frac{3}{4x^2+4}dx\). To rewrite it in a form reminiscent of the derivative of an arctangent, we can factor out the 4 in the denominator. Then the integral becomes \(\int \frac{3}{4(x^2+1)}dx\). We can finally factor out the 4 in the denominator which doesn't depend on the variable of integration, thus getting \(\frac{3}{4}\int \frac{1}{x^2+1}dx\). The function \(\frac{1}{x^2+1}\) is the derivative of \(\arctan(x)\). Thus, the antiderivative of our function will contain the function \(\arctan(x)\).

Step 3: Compute The Antiderivative

The antiderivative of \(\frac{1}{x^2+1}\) is \(\arctan(x)\). Thus, the antiderivative of our function is \(\frac{3}{4}\arctan(x)\). The most general antiderivative also contains a constant of integration C since a constant doesn't affect the derivative. Hence, the general antiderivative of our function is \(\frac{3}{4}\arctan(x) + C\).

Key Concepts

Understanding Integral Calculus

Integral calculus is a branch of mathematics focused on the concept of integration, which is essentially finding the antiderivative or the original function given its derivative. Integrals can be definite or indefinite. In this context, we're looking at indefinite integrals, resulting in a family of functions.

• **Antiderivative**: This is a function whose derivative is the given function. It is often accompanied by a constant of integration.
• **Indefinite Integrals**: Represented by the symbol \( \int \), these yield a general form of the antiderivative, including a constant \( C \).
• **Rational Fractions**: These involve expressions formed by dividing two polynomials, as seen in our example.

By rewriting expressions in the integral to match known derivatives, like the arctangent, we simplify the process of integration.

Exploring the Arctangent Function

The arctangent function, denoted as \( \arctan(x) \), is the inverse of the tangent function. It is a crucial tool in calculus, especially when dealing with integrals involving fractions similar to our exercise's form.

Key Properties

• **Derivative**: The derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \). This fact helps us recognize the integral \( \int \frac{1}{x^2+1} \, dx \) as related to the arctangent function.
• **Range**: Arctan values lie between \(-\frac{\pi}{2}\) and \( \frac{\pi}{2} \), making it a practical choice for integrations yielding outputs within this range.
• **Graph**: The graph of the arctangent function provides a smooth curve that approaches horizontal asymptotes, illustrating its bounded nature.

In our solution, recognizing the expression \( \frac{1}{x^2+1} \) as the derivative of the arctangent function allows us to determine the antiderivative efficiently.

The Constant of Integration

When finding antiderivatives, including the constant of integration \( C \) is imperative to capture the entire family of possible solutions.

Why is it Important?

• **Universality**: A derivative erases constant addition, so the antiderivative process must restore any unknown constant that was present initially.
• **Generality**: Including \( C \) ensures the solution reflects all possible functions whose derivative matches the integrand.
• **Problem Context**: Solving specific problems may require determining \( C \) based on initial conditions or additional information.

In our exercise, the general antiderivative \( \frac{3}{4} \arctan(x) + C \) reflects this principle, acknowledging the possible variations in solutions without specific constraints.