Problem 5 - Applied Calculus

Question

Find each indefinite integral. \(\int \sqrt{u} d u\)

Step-by-Step Solution

Answer

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

Step 1: Recognize the form of the integrand

The integrand is \( \sqrt{u} \), which can be rewritten using an exponent. Recognize that \( \sqrt{u} = u^{1/2} \). This helps us to apply the power rule for integration.

Step 2: Apply the power rule for integration

The power rule for integration states that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) where \( n eq -1 \). Here, \( n = \frac{1}{2} \), so apply the rule: \[ \int u^{1/2} \, du = \frac{u^{1/2 + 1}}{1/2 + 1} + C = \frac{u^{3/2}}{3/2} + C \]

Step 3: Simplify the expression

Simplify the expression \( \frac{u^{3/2}}{3/2} \) by multiplying by the reciprocal of the fraction: \[ \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \] Therefore, the integral becomes \( \frac{2}{3} u^{3/2} + C \).

Key Concepts

Power Rule for Integration

When dealing with integrals, one of the most essential tools is the power rule for integration. This rule can make the task of finding indefinite integrals significantly easier. The rule states:

• For any expression of the form \( u^n \), the integral is calculated as \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) where \( n eq -1 \).

To remember this, think of adding 1 to the exponent and then dividing by that new exponent value. Don’t forget the \( + C \) part, which represents the constant of integration. This constant is vital in indefinite integrals since they represent a family of functions.
This rule simplifies many integral problems because it converts the process into a straightforward calculation, as seen with the example \( \int u^{1/2} \, du \) becoming \( \frac{u^{3/2}}{3/2} + C \) after applying the power rule.

Integrand

The integrand is a crucial term in integration. It's the function you want to integrate. In simple terms, it's the 'inside' part of the integral that you apply the integration process to. When you see your expression, it generally looks like this: \( \int f(x) dx \). Here, \( f(x) \) is the integrand.
For the exercise, the integrand is \( \sqrt{u} \). Understanding the nature of the integrand helps in selecting the right method or rule to apply, such as possible simplifications or rewriting, as done by expressing \( \sqrt{u} \) as \( u^{1/2} \).
Recognizing patterns and forms in the integrand, like polynomial forms that suit the power rule for integration, can streamline the entire process of integration.

Exponentiation in Integrals

Exponentiation in the realm of integrals refers to the powers within the functions being integrated. This can appear intimidating initially, especially with roots or fractional exponents. However, rewriting expressions with exponents is a strategy to simplify integration.
For instance, rewrite \( \sqrt{u} \) as \( u^{1/2} \). This step might seem modest but is crucial as it allows for easy application of rules like the power rule.

• This makes manipulation and understanding of the expressions simpler.
• It gives a clear path to apply integration techniques.

Becoming comfortable with exponentiation—rewriting roots with fractional exponents or simplifying complex expressions—unlocks more straightforward approaches to integration problems and demystifies what can initially seem like a complex mathematical operation.