Problem 9 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int x \sqrt{x^{2}+1} d x$$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{3} (x^2 + 1)^{3/2} + C \).

Step 1: Choose a substitution

We start by choosing a substitution to simplify our integral. Let's take \( u = x^2 + 1 \). Then, \( du = 2x \, dx \) or \( \frac{1}{2} \, du = x \, dx \). This substitution simplifies the integral to \( \int \sqrt{u} \cdot \frac{1}{2} \, du \) or \( \frac{1}{2} \int u^{1/2} \, du \).

Step 2: Integrate the substitution

Now, we integrate \( \frac{1}{2} \int u^{1/2} \, du \). The integral of \( u^{1/2} \) with respect to \( u \) is \( \frac{2}{3} u^{3/2} \). Thus, our integral becomes \( \frac{1}{2} \times \frac{2}{3} u^{3/2} = \frac{1}{3} u^{3/2} + C \), where \( C \) is the constant of integration.

Step 3: Substitute back to x

Revert the substitution to express the integral in terms of \( x \). Recall \( u = x^2 + 1 \), so \( u^{3/2} = (x^2 + 1)^{3/2} \). Thus, our integral becomes \( \frac{1}{3} (x^2 + 1)^{3/2} + C \).

Step 4: Differentiate to check the solution

To verify our solution, differentiate \( \frac{1}{3} (x^2 + 1)^{3/2} + C \) with respect to \( x \). Using the chain rule: \[ \frac{d}{dx}\left( \frac{1}{3}(x^2 + 1)^{3/2} \right) = \frac{1}{3} \cdot \frac{3}{2} (x^2 + 1)^{1/2} \cdot 2x = x (x^2 + 1)^{1/2} \]. This matches the original integrand, confirming our solution is correct.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify integrals by introducing a new variable. It involves replacing a complex expression with a simpler one, making the integral easier to solve. In the given exercise, we used a substitution to tackle the integral \( \int x \sqrt{x^{2}+1} \, dx \). The key here is to identify a suitable substitution.

A good choice is often a function inside a square root or raised to a power, as these can complicate direct integration. We chose \( u = x^2 + 1 \), a common substitution when dealing with expressions involving \( x^2 \) and constants.

This choice helps us transform the original integral into a more straightforward form:

• Start with \( u = x^2 + 1 \)
• Differentiate to get \( du = 2x \, dx \)
• Replace \( x \, dx \) with \( \frac{1}{2} \, du \)

These steps substituted and simplified the integral to \( \frac{1}{2} \int u^{1/2} \, du \). Helpful, right? It's about easing the complexity, one step at a time.

Differentiation

To ensure our integration is correct, differentiation serves as a verification tool. Once we've solved the integral, it's wise to differentiate the result to see if it equals the original integrand. Let’s dive into this step.

After solving the integral and reverting our substitution, we found the result as \( \frac{1}{3} (x^2 + 1)^{3/2} + C \). Differentiation involves applying the chain rule—a derivative rule particularly useful here—to bring us back to our original expression.

Here’s what happens:

• Start by differentiating \( \frac{1}{3} (x^2 + 1)^{3/2} \)
• Apply the chain rule: \( \frac{d}{dx}(a(u(x))) = a'(u(x)) \cdot u'(x) \)

This gives us \( x (x^2 + 1)^{1/2} \), mirroring the integrand we began with. Ensuring the differentiated result matches the original integrand confirms the correctness of our integration process.

Chain Rule

The chain rule is indispensable in both integration and differentiation, especially when differentiating composite functions. In this exercise, it played a crucial role in confirming the validity of our integrated solution.

So, what is the chain rule? It tells us how to differentiate a function within a function, which is exactly what we encountered here.

• The outside function: \( a(x) = (x^2 + 1)^{3/2} \)
• The inside function: something like \( b(x) = x^2 + 1 \)
• The chain rule formula: \( rac{d}{dx} (a(b(x))) = a'(b(x)) \cdot b'(x) \)

Applying it meticulously, we took the derivative of the outer function while keeping the inner one constant, then multiplied by the derivative of the inner function.
This showed us that understanding composition in functions is key, allowing us to seamlessly transition back to the original function we integrated.

Constant of Integration

Ever noticed the plus \( C \) at the end of indefinite integrals? The "constant of integration" is a vital piece whenever we integrate.

Why is it so important? Indefinite integration involves finding a family of functions that differ only by a constant. Since differentiation of a constant is zero, this constant is needed to account for all possible antiderivatives of a function.

Think back to our integration result, \( \frac{1}{3} (x^2 + 1)^{3/2} + C \). Here, \( C \) represents any constant value since adding a constant doesn't affect the derivative.

• Integrals involve finding general solutions.
• The constant of integration \( C \) ensures completeness.

In practical applications, if given initial conditions, \( C \) can be specified. But in indefinite cases like our problem, it simply remains as \( C \), adding to the versatility and generality of our solution.