# How do you integrate #int x(5^(-x^2))dx#?

The answer is

We use the substitution

Therefore,

Then taking the logarithm

Therefore,

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To integrate (\int x \cdot 5^{-x^2} , dx), we can use the technique of substitution. Let (u = -x^2), then (du = -2x , dx). Solving for (dx), we get (dx = -\frac{du}{2x}). Substituting these into the integral, we have (\int x \cdot 5^{-x^2} , dx = -\frac{1}{2} \int e^u , du). Now, integrate (-\frac{1}{2} \int e^u , du) with respect to (u). After integrating, substitute back (u = -x^2) to obtain the final answer.

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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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