Okay, so I need to find the integer value of x in the arithmetic sequence 3², x, 3⁴. Let me start by recalling what an arithmetic sequence is. In an arithmetic sequence, the difference between consecutive terms is constant. That means the difference between the first and second term is the same as the difference between the second and third term. 

First, let me calculate the first term. 3 squared is 9, so the first term is 9. The third term is 3 to the power of 4. Let me compute that. 3⁴ is 3*3*3*3, which is 81. So the sequence is 9, x, 81. 

Since it's an arithmetic sequence, the difference between the second and first term should be equal to the difference between the third and second term. Let me denote the common difference as d. Then, according to the arithmetic sequence property:

x - 9 = 81 - x

Now, I need to solve for x. Let me write that equation down:

x - 9 = 81 - x

To solve for x, I can add x to both sides to get the x terms on one side. That would give:

x + x - 9 = 81

Which simplifies to:

2x - 9 = 81

Then, I can add 9 to both sides to isolate the term with x:

2x = 81 + 9

2x = 90

Now, divide both sides by 2:

x = 90 / 2

x = 45

So, the value of x should be 45. Let me check if this makes sense. The first term is 9, then 45, then 81. Let's compute the differences. The difference between 9 and 45 is 36, and the difference between 45 and 81 is also 36. Yep, that's a common difference, so it checks out. 

Wait, but the problem says the integer value of x. 45 is an integer, so that's good. There's no need to check for fractions or anything else. 

Just to make sure I didn't make any calculation errors. Let me recalculate 3⁴. 3^1 is 3, 3^2 is 9, 3^3 is 27, 3^4 is 81. Correct. Then the equation x - 9 = 81 - x. Solving: 2x = 90, x = 45. Yes, that's right. 

Alternatively, maybe there's another way to approach this. For an arithmetic sequence with three terms, the middle term is the average of the first and third terms. So, x should be (9 + 81)/2. Let me compute that: 9 + 81 is 90, divided by 2 is 45. Yep, same result. That's another way to think about it. That confirms the answer again. 

So, both methods lead to x being 45. I think that's solid. No mistakes here. The answer is 45.
Therefore, the integer value of $x$ in the arithmetic sequence $3^2, x, 3^4$ is $\boxed{45}$.
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