Problem 23 - Calculus of a Single Variable

Question

Evaluating a Definite Integral In Exercises \(21-32\) evaluate the definite integral. $$ \int_{0}^{\sqrt{3} / 2} \frac{1}{1+4 x^{2}} d x $$

Step-by-Step Solution

Answer

The evaluated integral is \(\pi/6.\)

Step 1: Recognizing the integral

This problem involves recognizing that the integral \(\int \frac{1}{1+4x^2} dx\) has a standard result, which is \(\frac{1}{2}arctan(2x)\). The \(4x^2\) in the denominator should hint at a substitution with a trigonometric function.

Step 2: Applying integral limits

Using the antiderivative found, evaluate at the upper limit of integration and then the lower limit. These are \(\sqrt{3}/2\) and 0, respectively. We get \(\frac{1}{2}arctan(2*\sqrt{3}/2) - \frac{1}{2}arctan(2*0)\). This simplifies to \(\frac{1}{2}arctan(\sqrt{3}) - 0\)

Step 3: Simplifying further

Knowing that \(arctan(\sqrt{3})\) is equal to \(\pi/3\), substitute this into the equation to simplify it to: \(\frac{1}{2}*\pi/3 = \pi/6.\)

Key Concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is a fundamental concept in calculus. It reverses the process of differentiation to find a function whose derivative matches a given function. In other words, if given a derivative, the antiderivative helps retrieve the original function.

When evaluating definite integrals, like in the example provided, the antiderivative is used to determine the area under the curve between two points. For the given integral \( \int \frac{1}{1+4x^2} \, dx \), the antiderivative is \( \frac{1}{2} \arctan(2x) \). Recognizing this form is crucial because it simplifies the integration process.

• The process involves identifying the derivative within the integral.
• Once recognized, apply the antiderivative corresponding to this derivative form.
• Finally, use the antiderivative to solve definite integrals by evaluating it at the defined upper and lower limits.

Trigonometric Substitution

Trigonometric substitution is a powerful technique in calculus used for simplifying the integration process, especially when dealing with complex polynomials. It involves substituting a trigonometric function for a variable to simplify an integral's expression.

In the problem at hand, substituting noticed elements with trigonometric terms simplifies the integral \( \int \frac{1}{1+4x^2} \, dx \). The appearance of \(4x^2\) indicates a potential substitution using the trigonometric identity \( \tan(u) \), resulting in \(1 + (\tan(u))^2 = \sec^2(u)\).

• This substitution transforms the integral into a known standard form.
• It assists in recognizing antiderivative forms, such as the \( \arctan(u)\) function.

By applying trigonometric substitution, integrals that initially seem complex become easier to handle and solve.

Arctan Function

The \(\arctan\) function, or the inverse tangent function, is crucial in evaluating integrals involving specific polynomial forms. It returns the angle whose tangent is the given number.

In the presented problem, identifying the connection between the integrand \(\frac{1}{1+4x^2}\) and the \(\arctan\) function is essential. This identification leads to the antiderivative \(\frac{1}{2}\arctan(2x)\), which simplifies the calculation.

• The \(\arctan\) function is invaluable when working with integrals that resemble \(\frac{1}{a^2+x^2}\).
• The understanding of its properties helps evaluate definite integrals involving trigonometric substitutions efficiently.

Connecting the integral to \(\arctan\) offers straightforward simplification and solutions for otherwise intricate expressions.

Integral Limits

Definite integrals involve finding the area under a curve between specified bounds, known as limits. These limits transform the antiderivative into a numeric value representing this area.

With the integral \(\int_{0}^{\sqrt{3} / 2} \frac{1}{1+4x^{2}} \, dx \), evaluating at the upper limit, \(\sqrt{3}/2\), and lower limit, 0, provides the definite answer. This is calculated by substituting these values into the antiderivative:

\[ \frac{1}{2} \arctan\left( 2\cdot\frac{\sqrt{3}}{2} \right) - \frac{1}{2} \arctan(0) \]

This initializes as \( \frac{1}{2}\arctan(\sqrt{3}) - 0 \). Recognizing \( \arctan(\sqrt{3}) = \pi/3 \), the definite integral evaluates to \( \pi/6 \).

• Always substitute the limits into the antiderivative.
• Simplify the expression to calculate the exact area under the curve.

Using integral limits, numerical solutions from antiderivatives are both meaningful and practical.