Indefinite Integrals

In summary, an indefinite integral is an antiderivative of a function, represented by the symbol ∫. It differs from a definite integral in that it does not have upper and lower limits, representing a family of functions rather than a single value. The process for solving an indefinite integral involves using various rules and adding a constant of integration at the end. Indefinite integrals and derivatives are inverse operations, and not all functions have an indefinite integral, requiring the function to be continuous and differentiable on the interval being integrated. Some functions, such as trigonometric functions, have more complex antiderivatives that cannot be expressed in terms of elementary functions.
  • #1
courtrigrad
1,236
2
When they ask you to set up integral formulas for the derivative of 1/(x+1), would you use the fundamental theorem of calculus and set up a definite integral, or an indefinite integral. Can someone help me clarify this?

Thanks
 
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  • #2
Integral formulas are (almost to my knowledge) expressed in indefinite inegrals, since all you need is an antiderivative of the integrand.

i.e.
[tex]\int\frac{dx}{x}=\ln x+C[/tex]
 
  • #3
for your question! When setting up integral formulas for the derivative of 1/(x+1), it is important to first consider what information is being asked for. The fundamental theorem of calculus applies to definite integrals, which involve finding the area under a curve between two specific points. In this case, it does not seem like we are being asked for a specific area, but rather the general formula for the derivative of 1/(x+1).

Therefore, it would be more appropriate to use an indefinite integral. An indefinite integral does not have specific limits of integration, but rather represents the antiderivative of a function. In this case, we can use the power rule for integration to find the antiderivative of 1/(x+1), which would give us ln(x+1) + C.

In summary, when asked to set up integral formulas for the derivative of 1/(x+1), it would be more appropriate to use an indefinite integral rather than the fundamental theorem of calculus. I hope this helps clarify things for you!
 

1. What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the antiderivative of a function. It is denoted by the symbol ∫ and is used to find the original function when the derivative is given.

2. How is an indefinite integral different from a definite integral?

An indefinite integral does not have upper and lower limits, whereas a definite integral does. This means that an indefinite integral represents a family of functions, while a definite integral represents a single value.

3. What is the process for solving an indefinite integral?

The process for solving an indefinite integral involves using the power rule, product rule, quotient rule, and chain rule to find the antiderivative of the function. The constant of integration, denoted by "C", is added at the end to represent the family of functions.

4. How do indefinite integrals relate to derivatives?

Indefinite integrals and derivatives are inverse operations of each other. This means that the derivative of an indefinite integral is equal to the original function, and the antiderivative of a derivative is equal to the original function multiplied by a constant.

5. Can all functions have an indefinite integral?

No, not all functions have an indefinite integral. For a function to have an indefinite integral, it must be continuous and differentiable on the interval being integrated. Additionally, some functions, such as trigonometric functions, have more complex antiderivatives that cannot be expressed in terms of elementary functions.

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