# Integration of bases other than e

• kari82
In summary, some common bases used in integration are 10, 2, and π. To integrate a function with a base other than e, one can use the substitution method or apply the appropriate rules for integration. An example of integrating a function with a base other than e is f(x) = 3x^2 with a base of 2, resulting in 3/ln(2) x^3 + C. E is often the preferred base for integration due to its useful properties, but there are limitations to integrating with bases other than e as not all functions can be integrated using them. In some cases, using e or a combination of bases may be necessary for proper integration.
kari82
Find the integral

∫(3^(2x))/(1 + 3^(2x)) dx

so I have that u= 1+3^(2x) and du=2ln3[3^(2x)]

I=1/(2ln3)∫3^(2x)(2)(ln3)/(1 + 3^(2x)) dx

Can anyone help me solve this please?

The point of the u-substitution is so that you make the change of variables and have an easier integrand to work with. Thus you actually need to substitute the u's in. For the denominator of the original integrand, this is clear (look at what substitution you made). Now du = 2 ln(3)[3^(2x)] dx (you need to remember this dx), so do you see how to take care of the numerator now?

## 1. What are some common bases used in integration besides e?

Some common bases used in integration are 10, 2, and π.

## 2. How do you integrate a function with a base other than e?

To integrate a function with a base other than e, you can use the substitution method or apply the appropriate rules for integration, such as the power rule or the chain rule.

## 3. Can you give an example of integrating a function with a base other than e?

Sure, an example would be integrating the function f(x) = 3x^2 with a base of 2. This would result in the integral of f(x) = 3x^2 dx being equal to 3/ln(2) x^3 + C.

## 4. Why is e often the preferred base for integration?

E is often the preferred base for integration because it has many useful properties, such as being the base for natural logarithms and having a derivative that is equal to itself. This makes it easier to integrate certain functions.

## 5. Are there any limitations to integrating with bases other than e?

There are some limitations to integrating with bases other than e, as not all functions can be integrated using these bases. In some cases, using e or a combination of bases may be necessary to properly integrate a function.

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