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Chapter 5: Problem 19

Express as one integral. $$ \int_{c}^{d} f(x) d x+\int_{e}^{c} f(x) d x $$

Short Answer

Expert verified
\( \int_{e}^{d} f(x) \).

Step by step solution

01

Reverse the Limit in the Second Integral

When we add two integrals, the limits must align properly. Notice that the second integral has the limits reversed. We can switch the limits on \( \int_{e}^{c} f(x) \) by introducing a negative sign: \[ \int_{e}^{c} f(x) \, dx = -\int_{c}^{e} f(x) \, dx. \] Now the expression becomes: \[ \int_{c}^{d} f(x) \, dx - \int_{c}^{e} f(x) \, dx. \]
02

Combine the Integrals

We can now rewrite both integrals with the same lower limit. This allows us to combine them into a single integral using properties of integrals. As a rule, \( \int_{a}^{b}f(x) \, dx + \int_{b}^{c}f(x) \, dx = \int_{a}^{c}f(x) \, dx \). The expression changes as follows: \[ \int_{c}^{d} f(x) \, dx - \int_{c}^{e} f(x) \, dx = \int_{c}^{d} f(x) \, dx + \int_{c}^{e} (-f(x)) \, dx = \int_{c}^{d} f(x) - \int_{c}^{e} f(x). \] Use the integral property \( \int f(x) - g(x) \; dx = \int \left(f(x) - g(x)\right) \, dx \) to express it as: \[ \int_{c}^{d}f(x) - \int_{c}^{e} f(x) = \int_{c}^{e} f(x) + \int_{e}^{d}f(x), \] which simplifies ultimately to: \[ \int_{e}^{d}f(x)\, dx. \]

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Key Concepts

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

Integral Properties
Integrals have several important properties that allow us to manipulate and combine them effectively. One of these properties is that reversing the limits of integration changes the sign of the integral. For instance, if you have an integral
  • \( \int_{a}^{b} f(x) \, dx \), which becomes \( -\int_{b}^{a} f(x) \, dx \).
This property helps when the limits of integration are not in the desired order.
Another useful property is the ability to add or subtract integrals with the same integrand and different intervals by adjusting the limits correctly. For example:
  • \( \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx \).
These concepts are crucial when simplifying expressions or solving integration problems.
Changing Limits of Integration
Sometimes you'll encounter integrals where the limits are reversed or not in an order that's easy to combine. You can alter these limits to make the integration process smoother. Let's say we're working with an integral \( \int_{e}^{c} f(x) \, dx \). To switch the limits from \( e \) to \( c \) to \( c \) to \( e \), you introduce a negative sign in front of the integral:
  • This means \( \int_{e}^{c} f(x) \, dx = -\int_{c}^{e} f(x) \, dx \).
This approach enables you to reorder the limits which can then help in combining with other integrals more seamlessly. By implementing this technique, you align the boundaries of your integrals, facilitating smoother operations and simplifications.
Combining Integrals
The act of combining multiple integrals into one can simplify and solve integration problems more efficiently. Once the limits have been adjusted to match, these integrals can be combined using basic properties and mathematical rules.
Assume we have:
  • \( \int_{c}^{d} f(x) \, dx \) and \(-\int_{c}^{e} f(x) \, dx \).
  • Using the property that \( \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx \), they become: \( \int_{e}^{d} f(x) \, dx \).
This transformation simplifies the original two integrals into a single integral \( \int_{e}^{d} f(x) \, dx \), which is often easier to solve or analyze. Thus, recognizing how to combine integrals effectively is an essential skill in calculus.

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Most popular questions from this chapter

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