Problem 57 - Thomas Calculus

Question

Evaluate the integrals. \(\int 3 x^{\sqrt{3}} d x\)

Step-by-Step Solution

Answer

\( \int 3x^{\sqrt{3}} \, dx = \frac{3}{\sqrt{3} + 1}x^{\sqrt{3} + 1} + C \).

Step 1: Identify the Integral Form

The integral given is \( \int 3x^{\sqrt{3}} \, dx \). This is a power form integral, where the integrand is \( 3x^{\sqrt{3}} \). We will apply the power rule for integration to solve this.

Step 2: Apply the Power Rule for Integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). Here \( n = \sqrt{3} \), so we integrate by adding 1 to the exponent and dividing by the new exponent.

Step 3: Integrate the Function

Using the power rule, add 1 to the exponent: \( \sqrt{3} + 1 = \sqrt{3} + 1 \). Now divide \( x^{\sqrt{3}+1} \) by \( \sqrt{3}+1 \), integrating the function. Include the constant coefficient of 3 in the integral. Combine terms to get:\[ \int 3x^{\sqrt{3}} \ dx = \frac{3}{\sqrt{3} + 1}x^{\sqrt{3} + 1} + C \]

Step 4: Simplify the Result

Simplifying the result: Finalize the expression:\[ \int 3x^{\sqrt{3}} \ dx = \frac{3}{\sqrt{3} + 1}x^{\sqrt{3} + 1} + C \] This is the indefinite integral of the function. The constant \( C \) is added because it represents the constant of integration in indefinite integrals.

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a key principle when dealing with integrals of power functions. It simplifies many problems involving finding antiderivatives. The rule provides a straightforward method to integrate functions of the form \( x^n \). Here's how it works:

• Consider a function \( x^n \). To integrate it, increase the exponent by one, turning \( n \) into \( n+1 \).
• Divide by the new exponent \( n+1 \), resulting in \( \frac{x^{n+1}}{n+1} \).
• Don't forget to add the constant of integration \( C \). This ensures that all potential antiderivatives are considered.

For example, if you have \( \int x^{\sqrt{3}} \, dx \), applying the power rule means:
1. Increasing the exponent: \( \sqrt{3} + 1 \)
2. Dividing by the new exponent: \( \frac{x^{\sqrt{3}+1}}{\sqrt{3}+1} \)
3. Including \( C \) for the solution's completeness. This makes the power rule a powerful and essential tool in calculus, especially for indefinite integrals.

Indefinite Integrals

Indefinite integrals represent a family of functions and are central in calculus. They are also known as antiderivatives. When you compute an indefinite integral, you're finding a function whose derivative equals the integrand.
Unlike definite integrals, indefinite integrals do not have upper and lower bounds. Instead, they account for all possible antiderivatives by adding a constant \( C \). This is often called the 'constant of integration.'
Indefinite integrals are essential in solving problems involving:

• Motion
• Area calculations
• Other physical and mathematical applications

Consider the integral \( \int 3x^{\sqrt{3}} \, dx \), which is an indefinite integral:

• The result must include a constant \( C \) since any constant derivative is zero.
• This means a family of functions can each serve as a solution.

By understanding indefinite integrals, you can better understand how functions change and behave over different regions of input values.

Exponents in Integrals

Exponents in integrals can change the approach to finding an antiderivative. Typical integrals involve constant exponents, but you can encounter fractional or irrational exponents as well. These require careful application of integration rules.
What happens when exponents are involved in integration?

• Normal rules apply: use the Power Rule for Integration.
• Adjust the exponent during integration by incrementing it by 1.
• Divide by this adjusted exponent to correctly balance the formula.

For the function \( 3x^{\sqrt{3}} \) in our integral, the exponent is \( \sqrt{3} \):

• The integration step involves adding 1 to \( \sqrt{3} \) and dividing by this sum.
• The challenge comes in calculating new expressions like \( \sqrt{3} + 1 \) precisely.
• Despite the complexity, the approach remains consistent whether dealing with simple integers, fractions, or more exotic exponents.

Mastering exponents in integrals enriches your calculus toolkit, enabling you to tackle a broader range of problems with confidence.