Problem 415 - Calculus Volume 1

Question

For the following exercises, find the antiderivatives for the functions. $$\int \frac{x d x}{\sqrt{x^{2}+1}}$$

Step-by-Step Solution

Answer

The antiderivative is \( 2\sqrt{x^2 + 1} + C \).

Step 1: Use Substitution Method

The integral \( \int \frac{x \, dx}{\sqrt{x^2 + 1}} \) suggests using a substitution to simplify. Let's set \( u = x^2 + 1 \). Then, the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 2x \), or \( du = 2x \, dx \). To relate it to \( x \, dx \), rewrite as \( x \, dx = \frac{1}{2} du \). Substitute \( u \) and \( x \, dx \) into the integral, resulting in \( \int \frac{1}{\sqrt{u}} \cdot \frac{1}{2} du \), allowing us to take out constants.

Step 2: Simplify the Integral

After substitution, the integral becomes \( \frac{1}{2} \int u^{-\frac{1}{2}} du \). This is a basic power rule integral.

Step 3: Apply the Power Rule for Integration

The power rule for integration states that \( \int u^n du = \frac{u^{n+1}}{n+1} + C \), where \( n eq -1 \). Here \( n = -\frac{1}{2} \). Thus, \( \int u^{-\frac{1}{2}} du = \frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C = 2u^{\frac{1}{2}} + C \).

Step 4: Substitute Back for Original Variable

Substitute back \( u = x^2 + 1 \) into the integral to revert to the variable \( x \). Thus, \( 2u^{\frac{1}{2}} + C \) becomes \( 2(x^2+1)^{\frac{1}{2}} + C \).

Step 5: Simplified Final Answer

So, the antiderivative \( \int \frac{x \, dx}{\sqrt{x^2 + 1}} = 2\sqrt{x^2+1} + C \), where \( C \) is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a powerful technique to simplify integral expressions, especially when the direct antiderivative is not readily apparent. In essence, it involves changing variables to transform a complex integral into a simpler form. Here's how it works: when given an integral like \( \int \frac{x \, dx}{\sqrt{x^2 + 1}} \), we look for a substitution that would streamline the process. A good substitution is often a function within the integrand, where its derivative is also present in the expression.

• Consider \( u = x^2 + 1 \). This choice is strategic since its derivative, \( du = 2x \, dx \), closely matches parts of our integral.
• The substitution rewrites the differential as \( x \, dx = \frac{1}{2} du \). This simplifies the integral to \( \int \frac{1}{\sqrt{u}} \cdot \frac{1}{2} du \), making it manageable to solve.

Returning to the original variable once the integration is complete is crucial to solving the problem. This involves substituting back the original expression for \( u \), leading us to the final solution in terms of \( x \). Remember, the key to substitution is to choose wisely and simplify your path to the antiderivative.

Power Rule for Integration

The power rule for integration is a foundational tool in calculus, letting you find antiderivatives of functions with ease. When applying the power rule, the form should generally be an algebraic expression like \( \int x^n \, dx \). The rule states that if \( n eq -1 \), then \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. This rule straightforwardly converts the integral of a power of \( x \) into a solvable expression.

• In the original problem, after substitution, our integral becomes \( \frac{1}{2} \int u^{-\frac{1}{2}} du \).
• Set \( n = -\frac{1}{2} \) in the rule, yielding \( \int u^{-\frac{1}{2}} du = \frac{u^{\frac{1}{2}}}{\frac{1}{2}} + C \).

After applying the power rule, you multiply by any constants pulled out during substitution. This is how the integral simplifies correctly, enabling efficient problem-solving.

Constant of Integration

The constant of integration, often denoted as \( C \), is a fundamental component in indefinite integrals. This constant accounts for any constant term that could be part of the original function's antiderivative. Since differentiation of a constant is zero, antiderivatives are unique up to a constant.

• When you find an antiderivative like \( \int \frac{x \, dx}{\sqrt{x^2 + 1}} = 2\sqrt{x^2 + 1} + C \), typical practice involves adding \( C \) to your solution.
• This ensures that all potential functions differing by a constant are represented.

In practical terms, without \( C \), your solution assumes a specific form, neglecting the infinite number of alternate forms that could exist for any antiderivative. Always remember to include \( C \) in your results to maintain mathematical accuracy and reflect the general solution to the integral problem.