# How do you find the indefinite integral of #int x(5^(-x^2))#?

The answer is

We do the integral by substitution

Therefore,

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To find the indefinite integral of ( \int x \cdot 5^{-x^2} ), we can use integration by substitution.

Let ( u = -x^2 ). Then, ( du = -2x , dx ).

Rewriting the integral with respect to ( u ):

[ \int x \cdot 5^{-x^2} , dx = -\frac{1}{2} \int 5^u , du ]

This becomes a straightforward integral of a constant base raised to the power of a variable:

[ = -\frac{1}{2} \cdot \frac{5^u}{\ln(5)} + C ]

Substitute back ( u = -x^2 ) into the equation:

[ = -\frac{1}{2} \cdot \frac{5^{-x^2}}{\ln(5)} + C ]

So, the indefinite integral of ( \int x \cdot 5^{-x^2} ) is ( -\frac{1}{2} \cdot \frac{5^{-x^2}}{\ln(5)} + C ), where ( C ) is the constant of integration.

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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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