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Chapter 5: Problem 15

Find the indefinite integral and check your result by differentiation. $$ \int d u $$

Short Answer

Expert verified
The indefinite integral of \( du \) is \( u + C \).

Step by step solution

01

Find the integral

The task is to find the indefinite integral of \( du \). The indefinite integral of \( du \) or \(\int du \) is simply \( u + C \), where \( C \) is the constant of integration.
02

Validate the result

To validate the accuracy of the result from the integration process, differentiate the result. The derivative of \( u \) is \( du \), and the derivative of a constant is zero, so derivative of \( u + C \) is \( du \).

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Key Concepts

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

Differentiation
Differentiation is a key concept in calculus that involves finding the derivative of a function. The derivative measures how a function changes as its input changes.
For example, if the function describes the position of an object over time, the derivative will describe the object's velocity.
  • The process of differentiation involves applying certain rules to the function, such as the power rule, product rule, quotient rule, and chain rule.
  • Each rule helps to find the derivative based on specific characteristics of the function.
Differentiation can also be used to check the accuracy of an indefinite integral. If you take the derivative of an integral's result and end up back at the original function, your integration was likely done correctly.
In the given exercise, we differentiated the result of the integral, which was \( u + C \), and ended up back with \( du \), confirming the correctness of our solution.
Constant of Integration
When you find an indefinite integral, you encounter the concept of the constant of integration, denoted as \( C \). This constant accounts for all the possible vertical shifts on the graph of an anti-derivative.
Remember:
  • The integral gives a family of functions, all differing by a constant value.
  • Without limits of integration, a specific integral value isn't defined, hence the inclusion of \( C \).
For instance, the indefinite integral \( \int du \) results in \( u + C \).
The function could be \( y = u + 0 \), \( y = u + 1 \), or even \( y = u + 7 \), and they would all still be valid primitives of the given derivative \( du \). This variability is solved when particular conditions, or initial values, are provided.
Basic Integration Rules
Basic integration rules are foundational to solving indefinite integrals efficiently. These rules are the counterparts to differentiation rules, making calculus a cohesive system.
  • One of the simplest rules is the integral of \( du \), which directly gives \( u + C \).
  • Additionally, the power rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
  • There are also rules for integrating exponential, logarithmic, trigonometric, and more complex functions.
Applying these rules helps simplify integrals without having to go through the entire derivation process each time.
In the provided exercise, using the rule for integrating a constant function \( du \) shows how straightforward integration can be. This is an example of how mastering these basic rules can significantly ease the calculation of integrals across a wide variety of functions.

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Most popular questions from this chapter

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