Problem 59

Question

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x $$

Step-by-Step Solution

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Answer

The definite integral evaluates to $ \frac{\pi}{4} $.

1Step 1: Identify the Integral

The given integral is $ \int_{0}^{1} \frac{1}{1+x^2} \, dx $. It involves the function $ f(x) = \frac{1}{1+x^2} $, which is similar to the derivative of the inverse tangent function.

2Step 2: Identify the Substitution

Notice that the derivative of $ \tan^{-1}(x) $ is $ \frac{1}{1+x^2} $. Thus, we can let $ u = \tan^{-1}(x) $. But since this is the integrand, it is also worth noting that the substitution can appear naturally.

3Step 3: Recognize Standard Integral

Recognize that $ \frac{1}{1+x^2} $ is the standard form for the derivative of the inverse tangent function: $ \frac{d}{dx} [\tan^{-1}(x)] = \frac{1}{1+x^2} $.

4Step 4: Apply Fundamental Theorem of Calculus

Since $ \frac{1}{1+x^2} $ is the derivative of $ \tan^{-1}(x) $, the integral becomes $ \tan^{-1}(x) $ evaluated from the limits 0 to 1.

5Step 5: Evaluate the Integral

Calculate $ \tan^{-1}(1) - \tan^{-1}(0) $. Since $ \tan^{-1}(1) = \frac{\pi}{4} $ and $ \tan^{-1}(0) = 0 $, the result is $ \frac{\pi}{4} - 0 $.

Key Concepts

Substitution RuleFundamental Theorem of CalculusInverse Trigonometric FunctionsAntiderivatives

Substitution Rule

The Substitution Rule is a powerful technique in calculus used to simplify integrals. It is particularly useful when dealing with complex integrals that can be transformed into more manageable ones.

By substituting a new variable for an expression in the integrand, integrals can often be rewritten into a simpler form.
This technique is similar to the chain rule in differentiation, but in reverse.

In this exercise, we did not perform a technical substitution since the integrand directly matched the derivative of an inverse trigonometric function. Nevertheless, recognizing patterns that allow for substitution is an essential skill. Identifying that the integral of the function $ \frac{1}{1+x^2} $ relates to the $ \tan^{-1}(x) $ function helps solve the integral without complications.
The key is to look for familiar derivative forms and understand how a substitution could simplify the integration process.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It is a crucial principle that makes evaluating definite integrals possible in a straightforward way.

It states that if a function is continuous on the closed interval $[a, b]$, and $F $ is an antiderivative of the function, then the integral from $a $ to $b $ of the function is $F(b) - F(a)$.

In the given problem, the function $ \frac{1}{1+x^2} $ has an antiderivative $ \tan^{-1}(x) $.
According to the Fundamental Theorem, the definite integral $ \int_0^1 \frac{1}{1+x^2} \, dx $ is simply $ \tan^{-1}(1) - \tan^{-1}(0) $.
Understanding this theorem allows us to transition smoothly from derivatives back to definite integrals.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when given the values of trigonometric functions.
These functions include $ \sin^{-1}(x) $, $ \cos^{-1}(x) $, and $ \tan^{-1}(x) $. Each function gives the angle whose trigonometric function results in a particular value.

The inverse tangent function, denoted as $ \tan^{-1}(x) $, is relevant in this exercise as its derivative is $ \frac{1}{1+x^2} $.

This property is vital in solving integrals like $ \int \frac{1}{1+x^2} \, dx $, since identifying its derivative helps us find the required antiderivative by reversing differentiation.
Thus, inverse trigonometric functions connect angles with their ratios, and their derivatives lay the groundwork for integration, as seen in this exercise.

Antiderivatives

Antiderivatives, also known as indefinite integrals, reverse the process of differentiation. Finding an antiderivative means finding a function whose derivative matches the given function.
In calculus, this process is central to solving definite integrals.

The antiderivative of $ \frac{1}{1+x^2} $ is $ \tan^{-1}(x) $, which was crucial for solving the integral in this problem.

Knowing and identifying standard antiderivatives helps in many integration problems.
For instance, understanding that the derivative of a function is $ \frac{dy}{dx} $ allows us to work backwards to find the original function from which it was derived.
This task exemplifies the importance of recognizing standard derivative forms to effectively determine their antiderivatives.

Problem 59

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