Problem 293 - Calculus Volume 1

Question

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} \frac{x}{\sqrt{1+x^{2}}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \sqrt{2} - 1 \).

Step 1: Identify the substitution

To solve the integral \( \int_{0}^{1} \frac{x}{\sqrt{1+x^{2}}} \, dx \), we will use a substitution method. Notice that the denominator involves \( 1+x^2 \), suggesting we can use \( u = 1 + x^2 \). This simplifies the square root function and its derivative can be easily matched with parts of the integrand.

Step 2: Calculate the derivative and differential

Differentiate \( u = 1 + x^2 \) to get \( \frac{du}{dx} = 2x \). Solving for \( dx \), we obtain \( dx = \frac{du}{2x} \). This substitute will help in changing the variable of integration from \( x \) to \( u \).

Step 3: Change the limits of integration

The original limits are from \( x = 0 \) to \( x = 1 \). Use the substitution to find the new limits. When \( x = 0 \), \( u = 1 + 0^2 = 1 \). When \( x = 1 \), \( u = 1 + 1^2 = 2 \). So, the limits change to \( u \) from 1 to 2.

Step 4: Substitute and simplify the integral

Substitute \( u = 1 + x^2 \) and \( dx = \frac{du}{2x} \) into the integral: \[ \int_{1}^{2} \frac{x}{\sqrt{u}} \cdot \frac{du}{2x}. \] The \( x \) terms cancel out, simplifying to \[ \frac{1}{2} \int_{1}^{2} \frac{1}{\sqrt{u}} \, du. \]

Step 5: Evaluate the integral

Now, solve \( \frac{1}{2} \int_{1}^{2} u^{-1/2} \, du \). The antiderivative of \( u^{-1/2} \) is \( 2u^{1/2} \). So \( \frac{1}{2} \cdot 2u^{1/2} = u^{1/2} \). Evaluate this from 1 to 2: \[ \left[ u^{1/2} \right]_{1}^{2} = 2^{1/2} - 1^{1/2}. \]

Step 6: Simplify the final answer

Compute the expression \( 2^{1/2} - 1^{1/2} = \sqrt{2} - 1 \). This is the value of the integral after evaluating it from 1 to 2.

Key Concepts

definite integrals

Definite integrals represent the area under a curve between two specific points, on a graph. In the integral notation \( \int_a^b f(x) \, dx \), \( a \) and \( b \) are the limits of integration. These limits define the interval on the x-axis over which the function \( f(x) \) is integrated.

The main idea is to calculate the exact area bounded by the function and the x-axis between these two points. Because definite integrals provide an area, they have a specific numerical value, unlike indefinite integrals that have a constant of integration.

A few important points about definite integrals:

• The value of a definite integral could be positive, zero, or negative, depending on the function's position relative to the x-axis.
• They do not include the constant of integration \( C \) since they result in a specific numeric value rather than a general form.
• In practice, they often require special integration techniques, such as substitution or integration by parts, to solve.

substitution method

The substitution method is a common technique used in integration to simplify the evaluation of integrals. This technique involves changing variables to transform a complex integral into a simpler one.

The purpose is to find a suitable substitution that simplifies the integrand, making it easier to integrate. Here’s how it is typically done:

• Identify a part of the integrand that can be rewritten using a new variable (often denoted \( u \)), simplifying its differentiation.
• Differentiate this new variable to determine \( du \), aiding in transforming the differential part of the integral \( dx \) in terms of \( du \).
• Rewrite the integrand in terms of \( u \) and \( du \) and then solve the simpler integral.

In our original exercise, we chose the substitution \( u = 1 + x^2 \) since the derivative \( 2x \) matches a part of the integrand. The substitution method simplifies solving the integral by removing complex expressions or inconvenient forms.

change of variables

The change of variables is a fundamental concept in calculus, particularly useful in integration. It allows us to modify the variable of integration, changing the appearance of the integrand to something more manageable.

When applying a change of variables:

• The original limits of integration are recalculated to fit the new variable. This involves substituting the limits through the chosen function.
• The integrand is rewritten in terms of the new variable, potentially simplifying the expression significantly.
• Solving the rewritten integral yields results that can be directly evaluated based on the new limits.

In this exercise, after choosing \( u = 1 + x^2 \), we recalculate the limits: originally from \( x = 0 \) to \( x = 1 \), which translates to \( u = 1 \) to \( u = 2 \). This new form often makes evaluating the integral more straightforward.

antiderivative

An antiderivative of a function is a function whose derivative is the original function. Finding antiderivatives is a crucial part of solving integrals, particularly indefinite integrals.

To solve an integral by finding its antiderivative, follow these steps:

• Understand the form of the original function, considering basic antiderivatives.
  
• Apply necessary integration techniques, possibly finding a substitution or using known identities.
• For definite integrals, calculate the antiderivative first, then compute its value at the various limits.

In our exercise, after making the variable change, we found the antiderivative of \( u^{-1/2} \), which is \( 2u^{1/2} \). This step transforms the integral into a direct evaluation task over the new limits, from \( u = 1 \) to \( u = 2 \). The difference of these evaluations gives the solution.