Are these two integral formulas the same?

Thread starter frasifrasi
Start date Sep 26, 2007
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Formulas Integral

In summary, the difference between two integral formulas may not be obvious at first and it is important to carefully examine their variables and limits. To determine if two formulas are equivalent, one can manipulate them or use integration techniques. While two different formulas can lead to the same result, they may still have subtle differences. It is important to understand the limitations of using the same formula in different contexts. To prove that two formulas are the same, one can use mathematical techniques, properties, or numerical methods.

Sep 26, 2007

frasifrasi

In my book, we were give:

int of tanx dx = - ln|cos x| + c

and later I cam across:

int of tanx dx = ln |sec x| + c

--should I use these interchangeably or what?

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Sep 26, 2007

D H

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What is the relation between [itex]\cos x[/itex] and [itex]\sec x[/itex]?

Sep 26, 2007

genneth

[tex]ln\left(\frac{1}{x}\right) = -ln x \quad \textrm{and} \quad \sec x = \frac{1}{\cos x}[/tex]

So yes, you can use them interchangeably.

FAQ: Are these two integral formulas the same?

What is the difference between the two integral formulas?

The two integral formulas may look different at first glance, but they may actually be equivalent or represent the same mathematical concept. It is important to carefully examine the variables and limits in each formula to determine their similarities and differences.

How can I tell if two integral formulas are the same?

To determine if two integral formulas are the same, you can try to manipulate one formula to make it look like the other. This can involve changing the limits, substituting variables, or using integration techniques such as integration by parts or trigonometric identities. If the resulting formulas are identical, then the two original formulas are equivalent.

Can two different integral formulas lead to the same result?

Yes, it is possible for two different integral formulas to lead to the same numerical result or area under the curve. This is because there are multiple ways to write an integral and different integration techniques can yield the same solution. However, the two formulas may still have subtle differences in terms of their variables or limits.

Are there any limitations to using the same integral formula in different contexts?

While an integral formula may be mathematically equivalent, it may not always be applicable in different contexts. For example, a formula for calculating the area under a curve may not be suitable for finding the volume of a solid of revolution. It is important to understand the context and assumptions of a formula before using it in a different scenario.

How can I prove that two integral formulas are the same?

To prove that two integral formulas are the same, you can use mathematical techniques such as substitution, integration by parts, or trigonometric identities to transform one formula into the other. You can also use mathematical properties and theorems to show that the two formulas are equivalent. Alternatively, you can use numerical methods to evaluate both formulas and compare the results.