# How do you integrate #intx\(3^(x^2+1))dx#?

The answer is

Therefore,

Therefore,

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To integrate (\int 3^{x^2+1} dx), you can use a substitution method. Let (u = x^2 + 1). Then (du = 2x dx). Solving for (dx), we get (dx = \frac{1}{2x} du).

Substitute (u = x^2 + 1) and (dx = \frac{1}{2x} du) into the integral:

(\int 3^{x^2+1} dx = \int 3^u \cdot \frac{1}{2x} du)

Now, since (u = x^2 + 1), we need to express (x) in terms of (u). (x^2 = u - 1), so (x = \sqrt{u - 1}).

Substitute (x = \sqrt{u - 1}) into the integral:

(\int 3^u \cdot \frac{1}{2\sqrt{u-1}} du)

Now, this integral can be solved using standard techniques.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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