2 unit - Integration and the Logarithmic Function (1 Viewer)

[V_L]

Hi can someone please give me the method to solve the following:

Find the indefinite integral (primitive function) of 3^(2x − 1)

The answer is [3^(2x − 1)]/2loge3 + C

The textbook doesn't have a method and my teacher said that we don't really need to know how because it is extension work but i would like to know how )

 

[InteGrand]

Hi can someone please give me the method to solve the following:

Find the indefinite integral (primitive function) of 3^(2x − 1)

The answer is [3^(2x − 1)]/2loge3 + C

The textbook doesn't have a method and my teacher said that we don't really need to know how because it is extension work but i would like to know how )

 

[pikachu975]

Hi can someone please give me the method to solve the following:

Find the indefinite integral (primitive function) of 3^(2x − 1)

The answer is [3^(2x − 1)]/2loge3 + C

The textbook doesn't have a method and my teacher said that we don't really need to know how because it is extension work but i would like to know how )

Basically you convert it to something that you can integrate. You think:
x... no
trig... no
exponential... yes

So you make it e^ln(your function) because you know e^ln cancels out to equal the function. From there you apply basic log laws and integrate/simplify.

 

[He-Mann]

Here is another way to think about it.

Let y = 3^{2x − 1}.

You cannot differentiate this easily; the problem is the power.

To solve this problem, you take logs on both sides to 'bring down' the power: log(y) = (2x - 1)log(3).

Make y the subject: y = e^{(2x - 1)log(3)} and differentiate normally.

This way you don't need to have e^{ln z} = z readily in memory (in fact, we just proved it) but it is very useful to commit it memory.