Problem 17 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(6 \sqrt{x}+\frac{1}{\sqrt[3]{x}}\right) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \( 4x^{3/2} + \frac{3}{2}x^{2/3} + C \).

Step 1: Rewrite the integrand using exponent notation

Rewrite the terms inside the integral in terms of exponents. Remember that \( \sqrt{x} = x^{1/2} \) and \( \frac{1}{\sqrt[3]{x}} = x^{-1/3} \). Thus, the integral becomes \( \int \left( 6x^{1/2} + x^{-1/3} \right) dx \).

Step 2: Integrate each term separately

Apply the power rule of integration, which states \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. For the first term, \( \int 6x^{1/2} \) becomes \( 6 \cdot \frac{x^{3/2}}{3/2} = 4x^{3/2} \). For the second term, \( \int x^{-1/3} \) becomes \( \frac{x^{2/3}}{2/3} = \frac{3}{2} x^{2/3} \).

Step 3: Combine the integrated terms and add the constant of integration

Combine the results from each term: \( 4x^{3/2} + \frac{3}{2}x^{2/3} + C \), where \( C \) is the constant of integration.

Key Concepts

Integrand Rewriting

The process of integrand rewriting involves transforming the terms within an integral into a format that's easier to work with when integrating.This helps simplify complex expressions, allowing more straightforward application of integration rules.
When you see expressions such as square roots or fractions involving roots in the integrand, try expressing them using exponent notation.For example:

• The square root of a variable, like \( \sqrt{x} \), can be rewritten as \( x^{1/2} \).
• Similarly, for cube roots in the denominator, like \( \frac{1}{\sqrt[3]{x}} \), this can be rewritten as \( x^{-1/3} \).

Using exponent notation for these expressions provides a smoother transition to applying integration techniques, such as the power rule of integration.

Power Rule of Integration

The power rule of integration is one of the foundational techniques for finding antiderivatives in calculus.
It states that the indefinite integral of \( x^n \) with respect to \( x \) is given by \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \( n eq -1 \) and \( C \) is the constant of integration.
This rule provides a simple and direct way to integrate most polynomial functions.
Here's how it works:

• Identify the exponent \( n \) in your expression. This is the power of \( x \) in the term you need to integrate.
• Apply the formula: increment the exponent by 1 and divide by this new exponent.

Using this method makes integrating terms with exponents straightforward and efficient, especially when combined with previously rewritten exponents from integrand rewriting.

Exponent Notation

Exponent notation is a critical mathematical tool used not only in integration but across many areas of mathematics.This notation helps express roots and powers using exponents, which can simplify calculations and algebraic manipulations.
For example:

• \( x^{1/2} \) signifies the square root of \( x \).
• \( x^{1/3} \) signifies the cube root of \( x \).
• \( x^{-1/3} \) represents the reciprocal of the cube root of \( x \), equivalent to \( \frac{1}{x^{1/3}} \).

When dealing with integration, rewrite any roots or fractions as powers of \( x \) to streamline the application of the power rule.This simplification step not only aids in integrating but enhances understanding and manipulation of mathematical expressions.
Using exponent notation allows you to tackle integration problems with confidence and clarity.