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Chapter 5: Problem 5

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

Short Answer

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

Step by step solution

01

Identify the Form of the Integral

The given integral is \( \int \sqrt{u} \, du \). We recognize that \( \sqrt{u} \) can be rewritten as \( u^{1/2} \), which allows us to apply the power rule for integration.
02

Recall the Power Rule for Integration

The power rule states that for any variable \( x \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). In our case, \( n = 1/2 \).
03

Apply the Power Rule

Using the power rule, we integrate \( u^{1/2} \):\[ \int u^{1/2} \, du = \frac{u^{1/2 + 1}}{1/2 + 1} + C = \frac{u^{3/2}}{3/2} + C \]
04

Simplify the Result

To simplify \( \frac{u^{3/2}}{3/2} \), multiply by the reciprocal of \( \frac{3}{2} \):\[ \frac{2}{3} u^{3/2} + C \]Thus, the integral \( \int \sqrt{u} \, du \) simplifies to \( \frac{2}{3} u^{3/2} + C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

power rule for integration
The power rule for integration is one of the most fundamental concepts in calculus. It allows us to find the antiderivative, or indefinite integral, of functions of the form \( x^n \). When using the power rule, remember:
  • If \( n eq -1 \), the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \).
  • \( C \) is the constant of integration, representing the family of all possible solutions.

For the integral \( \int \sqrt{u} \, du \), recognize the square root as \( u^{1/2} \). This reformation makes the power rule applicable by identifying the exponent. Simply increase the exponent by 1 and divide by the new exponent. This rule provides a straightforward technique to tackle many basic integrals effectively.
integral calculus
Integral calculus deals with finding quantities when rates of change are known. It involves processes like integration, which is essentially the opposite of differentiation.
  • Indefinite integrals, unlike definite integrals, do not have limits of integration and include a constant \( C \).
  • They provide a general form of antiderivative for a given function.

The goal in problems like \( \int \sqrt{u} \, du \) is to determine the function whose derivative is \( \sqrt{u} \). By using integral calculus, understanding and controlling how variables affect functions becomes evident. It's a tool that forms the basis for many fields of science and engineering.
mathematical simplification
Mathematical simplification is crucial in calculus to make complex expressions more manageable. After applying the power rule to the integral of \( \sqrt{u} \) which gives us \( \frac{u^{3/2}}{3/2} + C \), simplification involves reducing fractions and expressions into simpler forms.
  • Instead of leaving fractions complex, we simplify by multiplying with the reciprocal.
  • For example, \( \frac{u^{3/2}}{3/2} \) simplifies to \( \frac{2}{3} u^{3/2} \) when multiplied by the reciprocal of \( \frac{3}{2} \).

Such steps are vital for achieving final solutions that are easy to interpret. Simplification often turns a mathematically correct answer into a neat and clean final expression that reveals more about the function's behavior.

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