Problem 27 Find antiderivatives of the give... [FREE SOLUTION]

Chapter 25: Problem 27

Find antiderivatives of the given functions. $$f(x)=x^{2}-4+x^{-2}$$

Short Answer

Expert verified

$ F(x) = \frac{x^3}{3} - 4x - x^{-1} + C $ is an antiderivative of $ f(x) = x^2 - 4 + x^{-2} $.

Step by step solution

Understand the Function

The function given is a combination of three power functions: $ f(x) = x^2 - 4 + x^{-2} $. We need to find antiderivatives for each term separately and then combine these results.

Use Antiderivative Rules

Recall the power rule for antiderivatives: $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $ for $ n eq -1 $. We'll apply this rule to each term.

Find Antiderivative of Each Term

For $ x^2 $, the antiderivative is $ \frac{x^{2+1}}{2+1} = \frac{x^3}{3} $.For $ -4 $, the antiderivative is $ -4x $ (since the antiderivative of a constant $ c $ is $ cx + C $).For $ x^{-2} $, the antiderivative is $ \frac{x^{-2+1}}{-2+1} = \frac{x^{-1}}{-1} = -x^{-1} $.

Combine the Antiderivatives

Combine the antiderivatives of each term:$ \int f(x) \, dx = \frac{x^3}{3} - 4x - x^{-1} + C $, where $ C $ is the constant of integration.

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Key Concepts

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

Power Rule for Antiderivatives

The power rule is a fundamental tool for finding antiderivatives in calculus. It allows us to find an antiderivative for any function expressed as a power of x, as long as the power is not -1. Here鈥檚 how it works:

If you have a function like $ x^n $, the antiderivative can be found using: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where $ C $ is the constant of integration, and $ n $ is any number except -1.
This rule simplifies the process of finding antiderivatives by giving us a straightforward formula.

For example, in the function $ x^2 $, $ n = 2 $, so using our power rule, we find the antiderivative to be $ \frac{x^{3}}{3} $. Applying a similar technique to $ x^{-2} $, we have $ n = -2 $, so the antiderivative is $ -x^{-1} $.

Remember, this rule does not work when $ n = -1 $, which is a peculiarity that requires a different approach, using the natural logarithm function.

Constant of Integration

Whenever you calculate an antiderivative, it's crucial to include a constant of integration, denoted by $ C $. But why do we need this constant term? Let鈥檚 explore:

Antiderivatives represent a whole family of functions. The constant $ C $ accounts for all possible vertical shifts of the antiderivative. It indicates that there are infinite solutions differing by this constant value.
In the function $ f(x) $, for instance, if we calculate $ \int (f(x)) $, it will add $ C $ to signal that each antiderivative is part of this larger set of solutions.

Therefore, when you find $ \int f(x) \, dx $ as $ \frac{x^{3}}{3} - 4x - x^{-1} + C $, the $ C $ encapsulates the idea that there are many valid antiderivatives for this integral, all differing only by a constant.

Step-by-Step Solution

Approaching calculus problems step-by-step can turn complex tasks into manageable parts. Let鈥檚 walk through finding the antiderivative for $ f(x) = x^2 - 4 + x^{-2} $ using this approach:

Step 1: Break down the function $ f(x) = x^2 - 4 + x^{-2} $ into its components and focus on each one individually.
Step 2: Apply the power rule for each term:
- For $ x^2 $: $ \int x^2 \, dx = \frac{x^3}{3} $.
- For the constant $ -4 $: Its antiderivative is $ -4x $ because the antiderivative of a constant $ c $ is $ cx + C $.
- For $ x^{-2} $: Using the power rule, we find $ \int x^{-2} \, dx = -x^{-1} $.

By systematically applying these steps, you can ensure accuracy and understand each part of the process. Finally, combine these results to get the complete antiderivative: \[ \int f(x) \, dx = \frac{x^3}{3} - 4x - x^{-1} + C \] This organized approach not only aids in understanding but also helps in cementing the problem-solving process.

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91影视

Find antiderivatives of the given functions. $$f(x)=x^{2}-4+x^{-2}$$

Short Answer

Step by step solution

Understand the Function

Use Antiderivative Rules

Find Antiderivative of Each Term

Combine the Antiderivatives

Key Concepts

Power Rule for Antiderivatives

Constant of Integration

Step-by-Step Solution

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