Problem 14 - Calculus and its applications

Question

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

Step-by-Step Solution

Answer

\( \int \sqrt{x} \, dx = \frac{2}{3}x^{3/2} + C \).

Step 1: Recognize the Integral Form

The integral given is \( \int \sqrt{x} \, dx \). Recognize that \( \sqrt{x} \) can be rewritten using exponent notation as \( x^{1/2} \). So the integral becomes \( \int x^{1/2} \, dx \).

Step 2: Apply the Power Rule for Integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = \frac{1}{2} \). So we apply the rule: \[ \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C. \]

Step 3: Simplify the Expression

Simplify \( \frac{x^{3/2}}{3/2} \). Divide by a fraction \( \frac{3}{2} \) which is equivalent to multiplying by its reciprocal \( \frac{2}{3} \): \[ \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}. \]

Step 4: Write the Final Answer

Include the constant of integration \( C \) to obtain the indefinite integral: \[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} + C. \]

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental tool when handling integrals involving powers of variables, such as quadratic or cubic expressions. It allows us to integrate functions of the form \(x^n\), where \(n\) is any real number except \(-1\). This is because if \(n = -1\), the integral becomes a natural logarithm, which is a special case. The rule is expressed as follows:

• \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)

Applying this rule requires adding 1 to the exponent of \(x\) and then dividing by the new exponent. This becomes particularly useful for variable expressions requiring fundamental integration, such as polynomials. Remember, always include the constant \(C\), which represents the family of all antiderivatives. This process effectively finds the area under the curve described by the function over an indefinite range, providing a wider understanding of how the function behaves.

Exponentiation of Square Roots

Understanding exponentiation of square roots is crucial for simplifying integral expressions. When faced with a square root, it's important to express it using fractional exponents to apply calculus operations like differentiation or integration. For example, \(\sqrt{x}\) is effectively \(x^{1/2}\). Rewriting the expression in this form makes it compatible with the power rule for integration.

• Square root \(\sqrt{x}\) becomes \(x^{1/2}\)
• Cube root \(\sqrt[3]{x}\) becomes \(x^{1/3}\)

These transformations are beneficial because they transform radicals into forms that are easier to manipulate and integrate. For instance, once you've rewritten \(\sqrt{x}\) as \(x^{1/2}\), you can easily apply the power rule to find its integral.

Constant of Integration

When you calculate indefinite integrals, incorporating a constant of integration is essential. This constant, represented by \(C\), accounts for all the possible antiderivatives of a function. Remember, indefinite integrals yield a family of curves that parallel each other with the same slope, differing only by a constant vertical shift.

• \(C\) is arbitrary—it can be any value.
• The presence of \(C\) reflects the general solution.
• It highlights the indefinite nature of integral notations, unlike definite integrals which yield specific numerical results.Indefinite integration without \(C\) would preclude capturing all possible outputs of an antiderivative. This is significant for interpreting real-world phenomena, where initial values or boundary conditions help determine the exact form of \(C\) in applications such as physics, engineering, and statistics.