Problem 48 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int x^3 \sqrt{x^2 + 1} \, dx \)

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{1}{2} \left( \frac{(x^2 + 1)^3}{3} - \frac{(x^2 + 1)^2}{2} \right) + C \).

Step 1: Choose Substitution

We begin by selecting a substitution that simplifies the integral. Here, let us set \( u = x^2 + 1 \). This choice simplifies the expression under the square root.

Step 2: Find the Differential

With the substitution \( u = x^2 + 1 \), we differentiate to find \( du \). So, \( du = 2x \, dx \). Therefore, \( x \, dx = \frac{1}{2} du \).

Step 3: Express x in terms of u

Since \( u = x^2 + 1 \), we need \( x^3 \) in terms of \( u \). We know \( x^2 = u - 1 \), so \( x = \sqrt{u - 1} \) and \( x^3 = (u - 1)^{3/2} \).

Step 4: Substitute into the Integral

Substitute \( u \), \( du \), and \( x \) back into the integral: \( \int x^3 \sqrt{x^2 + 1} \, dx = \int (u-1)^{3/2} \cdot \sqrt{u} \cdot \frac{1}{2} \, du \). Simplify to \( \frac{1}{2} \int (u-1)^{3/2} \cdot u^{1/2} \, du \).

Step 5: Simplify and Integrate

The integral \( \int (u-1)^{3/2} u^{1/2} \, du \) simplifies to \( \int (u^2 - u) \, du \). Expanding and combining like terms gives \( \int (u^{2} - u) \, du \).

Step 6: Perform the Integration

Integrate term by term: \( \int u^2 \, du = \frac{u^3}{3} \) and \( \int u \, du = \frac{u^2}{2} \). So the integral becomes \( \frac{1}{2} \left( \frac{u^3}{3} - \frac{u^2}{2} \right) + C \).

Step 7: Back Substitute for x

Recall \( u = x^2 + 1 \). Substitute back to get the original variable: \( \frac{1}{2} \left( \frac{(x^2 + 1)^3}{3} - \frac{(x^2 + 1)^2}{2} \right) + C \).

Key Concepts

Substitution Method

The substitution method is a powerful tool in integration that makes complex integrals more manageable. It works by transforming the variable of integration, allowing us to simplify the integral significantly. In this exercise, we perform substitution by setting \( u = x^2 + 1 \). This choice simplifies the square root present in the original expression.
When choosing a substitution, the goal is to replace the complex part of the integral to achieve a simpler expression. Here are a few tips for choosing the right substitution:

• Look for expressions inside a function (like a square root or exponent) that can be simplified.
• Consider expressions that have their derivatives present elsewhere in the integral.

Once you have your substitution, the next step is to find the differential (\( du \)) as we did in the original solution. This involves differentiating \( u \) with respect to \( x \), leading to \( du = 2x \, dx \). We then express \( x \, dx \) in terms of \( du \), creating a neat package for integration.

Differential Calculus

Differential calculus focuses on how functions change, which is a key element in the substitution process. It involves finding the derivative of a function, which tells us the rate at which one quantity changes with respect to another. This is crucial in our exercise for transforming expressions and differentials correctly.
In our example, we had the substitution \( u = x^2 + 1 \). The differential calculus step here involved finding \( du \). By differentiating \( u \) with respect to \( x \), we got \( du = 2x \, dx \). This transformation is vital because it allows us to rewrite and simplify the integral for computation.
Here's a simplified recap:

• Find the function you wish to substitute and differentiate it with respect to \( x \).
• Use the derivative to express \( dx \) (or the differential) in terms of \( du \).
• Utilize these expressions to transform and simplify the original integral.

A good grasp of derivatives is essential not only for substitution but for many areas of calculus.

Integration Techniques

Integration techniques are systematic ways to solve integrals, and they're essential for calculus studies. In this task, we used substitution as our technique, which simplified the integral before performing the standard integration.
After substitution, an important part of integration is performing the actual integral. This exercise required expanding and integrating the expression \( \int (u^2 - u) \, du \). Each term is integrated separately, using the power rule for integrals:
\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]
In this scenario, the integral becomes:

• \( \int u^2 \, du = \frac{u^3}{3} \)
• \( \int u \, du = \frac{u^2}{2} \)

Substituting these antiderivatives into the equation and reapplying the original variable gives us the solution. Remember to always simplify and back-substitute where necessary to finish the integral in terms of the original variable \( x \).
Mastering these techniques involves practice and understanding each methodological step in calculus integrations.