Problem 13 - Calculus and its applications

Question

Find each integral. $$ \int \sqrt[3]{x} d x $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Integral Using Exponents

First, let's rewrite the cube root of \(x\) as an exponent. The cube root of \(x\) can be written as \(x^{1/3}\). This allows us to express the integral as: \[\int x^{1/3} \, dx\]

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, \(n = \frac{1}{3}\). Applying the power rule gives:\[\int x^{1/3} \, dx = \frac{x^{1/3 + 1}}{1/3 + 1} + C\]

Step 3: Simplify the Expression

Add the exponents: \(1/3 + 1 = 4/3\). So the expression becomes:\[\frac{x^{4/3}}{4/3} + C\]To simplify, multiply by the reciprocal of \(4/3\), which is \(3/4\):\[\frac{3}{4} x^{4/3} + C\]

Step 4: Finalize the Solution

Thus, the integral of \(\sqrt[3]{x}\) is:\[\int \sqrt[3]{x} \, dx = \frac{3}{4} x^{4/3} + C\]

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a fundamental tool in integral calculus, essential for solving problems like the one provided. It states that if you have an integral of the form \[ \int x^n \, dx \]where \( n eq -1 \), you can compute it using \( \frac{x^{n+1}}{n+1} + C \). Here, \( C \) is the constant of integration, which accounts for any constant that could be added when differentiating to return the original function.

• Remember that the power rule can only be applied to functions where the exponent is not equal to -1.
• The process involves increasing the current exponent by 1, then dividing by the new exponent.

In our example, we have \( n = 1/3 \). By increasing \( 1/3 \) by 1, we get \( 4/3 \), and divide by this result, making our expression \( \frac{x^{4/3}}{4/3} + C \). The next steps will focus on simplifying this expression accurately.

Exponents in Integration

Exponents play a crucial role when performing integration, especially when dealing with roots. To apply the power rule, we rewrite roots as fractional exponents. For instance, a cube root can be expressed as a power of one third. For instance:

• The cube root of \( x \) is represented as \( x^{1/3} \).

This approach is incredibly helpful because it aligns the integral with the form required by the power rule (\( x^n \)), facilitating its application.Recognize that manipulating exponents often involves simple arithmetic like adding fractions. In this case, adding 1 to \( 1/3 \) gives \( 4/3 \), transitioning our expression into a simpler format more suitable for integration. Such transformations are foundational steps that simplify further calculus operations.

Simplification of Integrals

After using the power rule, simplifying the resulting expression is essential to solving the integral completely. Thus, once we obtain the expression:\[ \frac{x^{4/3}}{4/3} + C \]we need to revise it to a more standard form. This involves multiplying by the reciprocal of the denominator:

• The reciprocal of \( 4/3 \) is \( 3/4 \).

When you multiply \( \frac{x^{4/3}}{4/3} \) by \( 3/4 \), you effectively cancel out the fractional denominator:\[ \frac{3}{4} x^{4/3} + C \]This step ensures that the integral is not only correct but also presented in a "cleaner" and more comprehensible form. It is an important final task to refine your expression, crucial in making your final solution both precise and elegant.