Problem 56 - Calculus Single Variable

Question

Calculate the given integral. $$ \int \frac{2 x}{\sqrt{x^{4}+1}} d x $$

Step-by-Step Solution

Answer

The integral is \( 2\sqrt{x^4 + 1} + C \).

Step 1: Recognize the Substitution

To simplify the integral, use the substitution method. Notice that the expression under the square root, \(x^4 + 1\), suggests using a substitution to simplify it. Let \( u = x^4 + 1 \).

Step 2: Differentiate the Substitution

Differentiate the substitution we found: \( u = x^4 + 1 \). Differentiating both sides with respect to \( x \), we get \( \frac{du}{dx} = 4x^3 \), so \( du = 4x^3 \, dx \).

Step 3: Adjust the Differential for the Integral

In the integral \( \int \frac{2x}{\sqrt{x^4 + 1}} \, dx \), we see a \( 2x \, dx \) term. However, from the differentiation step, we need \( x^3 \, dx \). Notice that \( 2x \, dx \) can be rewritten by dividing the equation \( du = 4x^3 \, dx \) by \( 2x^2 \): \( x^2 \, du = 2x \, dx \).

Step 4: Integrate with the New Variables

Now substitute the terms in the integral: \( \int \frac{2x}{\sqrt{x^4 + 1}} \, dx = \int \frac{1}{\sqrt{u}} \, \frac{x^2}{x^2} \, du \). Simplifies to \( \int \frac{1}{\sqrt{u}} \, du \). This integral simplifies to \( 2\sqrt{u} + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back the Original Variable

Replace \( u \) with its original expression in terms of \( x \), which is \( x^4 + 1 \). Thus, the integral \( 2\sqrt{u} + C = 2\sqrt{x^4 + 1} + C \).

Key Concepts

Substitution Method

When tackling integrals, the substitution method is a powerful technique. It simplifies evaluating complex integrals by transforming them into more manageable forms. This technique involves substituting a part of the integral expression with a new variable — known as the substitution variable. In our example, the complex expression under the square root, \(x^4 + 1\), is replaced by \(u\). This simplifies our integral:

• Identify the substitution: Choose \(u = x^4 + 1\).
• Transform the differential: Differentiate to find \(du = 4x^3\, dx\).
• Use the substitution in the integral: Rewrite the integral in terms of \(u\).

By substituting, a challenging integral can become much simpler to solve.

Differentiation

Differentiation plays a crucial role in the substitution method, particularly when transitioning from one variable to another. In the process of substitution, you must differentiate the substitution expression to transition the differential element. For the substitution \(u = x^4 + 1\), you differentiate to find:

• \(\frac{du}{dx} = 4x^3\) which implies \(du = 4x^3\, dx\).

This differentiation lets you express the original differential \(dx\) in terms of \(du\), enabling a seamless switch of variables. Ensuring that your differentiation is accurate and correctly applied is essential. It sets the stage for simplifying and solving the integral.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is fundamental when dealing with integrals:

• Indefinite Integrals: The integral without specified limits of integration. The result always includes a constant of integration, \(C\). For instance, in our exercise, we ended up with \(2\sqrt{u} + C\), representing the family of all antiderivatives of the function.
• Definite Integrals: These have specific upper and lower limits, resulting in a numerical value. They don't include the constant of integration.

In our example, we were solving an indefinite integral, highlighting the importance of understanding that there are no limits involved, just the antiderivative.

Constant of Integration

The constant of integration, \(C\), is an important element in indefinite integrals. When you integrate a function without defined limits, the answer includes \(C\) to represent all possible vertical shifts of the antiderivative.

• It's essential to remember because any antiderivative of a function can be altered by adding a constant.
• This ensures that all potential solutions to an indefinite integral are accounted for.

When performing substitution or direct integration, always include \(C\) in your final result if you're dealing with an indefinite integral. Not including \(C\) would imply a specific solution, which might mislead further calculations or applications of the integral. In our context, after substituting back \(u = x^4 + 1\), the final expression includes \(2\sqrt{x^4 + 1} + C\), capturing the full range of antiderivatives.