Problem 49 - Thomas Calculus in SI Units

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int 3 x^{\sqrt{3}} d x$$

Step-by-Step Solution

Answer

The antiderivative is \(\frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1} + C\).

Step 1: Identify the Function to Integrate

The function to integrate is given as \(f(x) = 3x^{\sqrt{3}}\). Here, the variable \(x\) is raised to the power \(\sqrt{3}\), and it is multiplied by the constant 3.

Step 2: Apply the Power Rule of Integration

The power rule for integration states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\). Here, \(n\) is \(\sqrt{3}\). Apply this rule to integrate the given function:\[\int 3x^{\sqrt{3}} \, dx = 3 \cdot \frac{x^{\sqrt{3}+1}}{\sqrt{3}+1} + C\]

Step 3: Simplify the Result

Simplify the expression obtained from applying the power rule. The integral becomes:\[\int 3x^{\sqrt{3}} \, dx = \frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1} + C\]This is the most general antiderivative of the function.

Step 4: Verify by Differentiation

Differentiate the result obtained to check if it matches the original function. Differentiate \(\frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1} + C\):1. Find the derivative of \(\frac{3}{\sqrt{3}+1}x^{\sqrt{3}+1}\): \[\frac{d}{dx} \left( \frac{3}{\sqrt{3}+1}x^{\sqrt{3}+1} \right) = \frac{3}{\sqrt{3}+1} \cdot (\sqrt{3}+1)x^{\sqrt{3}} = 3x^{\sqrt{3}}\]2. The constant \(C\) differentiates to 0.The differentiation confirms the original function \(3x^{\sqrt{3}}\), verifying the solution is correct.

Key Concepts

power rule of integration

The power rule of integration is a fundamental principle in calculus used to find the integral of a function. It is especially useful when dealing with functions that have the form \(f(x) = ax^n\), where \(a\) is a constant and \(n\) is any real number except -1.

To apply this rule, follow these steps:

• Add 1 to the exponent \(n\) of \(x\).
• Divide by the new exponent \(n + 1\).
• Don't forget to include the constant of integration \(C\).

In our example, the function \(3x^{\sqrt{3}}\) is integrated by applying this rule, resulting in \(\frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1} + C\). This provides the most general form of the integral, which includes all possible antiderivatives differentiated by \(C\).

Remember, this rule cannot be used directly if \(n = -1\) because it would involve division by zero, leading to a different approach known as the natural logarithm rule of integration.

antiderivative

The antiderivative is another name for the indefinite integral of a function. It essentially reverses the process of differentiation.

When you find an antiderivative, you are looking for a function \(F(x)\) whose derivative is the original function \(f(x)\). This means if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\).

Consider our example, \(3x^{\sqrt{3}}\). By integrating it, we determine that \(\frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1} + C\) is a function whose derivative gives back \(3x^{\sqrt{3}}\).

Antiderivatives are crucial because they allow us to solve problems involving accumulation, such as finding areas under curves or solving differential equations. The constant \(C\) in the antiderivative accounts for all vertical shifts of the function, showing the infinite family of solutions possible.

differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function value changes as its input changes. It tells us the rate at which the function is changing at any point.

To verify the correctness of an antiderivative, you differentiate it. This means transforming the function back into its original form.

In our case, differentiate \(\frac{3x^{\sqrt{3}+1}}{\sqrt{3}+1} + C\) back to \(3x^{\sqrt{3}}\). Here's how that works:

• The term \(\frac{3}{\sqrt{3}+1}\) is a constant, so it is pulled out of the differentiation process.
• The exponent rule is applied, reducing the power of \(x\) by 1, resulting in \(3x^{\sqrt{3}}\).
• The constant \(C\) differentiates to 0 as it does not change.

Checking an integral using differentiation ensures that the antiderivative is correctly calculated, maintaining accuracy and reliability in solving calculus problems.