What is #int x*e^(x^2) dx# ?
So:
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To integrate ( x \cdot e^{x^2} ) with respect to ( x ), you can use substitution. Let ( u = x^2 ), then ( du = 2x , dx ). Rewriting the integral:
[ \int x \cdot e^{x^2} , dx = \frac{1}{2} \int e^u , du ]
Now, integrate ( e^u ) with respect to ( u ):
[ \int e^u , du = e^u + C ]
Substitute back ( u = x^2 ):
[ \frac{1}{2} \int e^{x^2} , dx = \frac{1}{2} e^{x^2} + C ]
So, ( \int x \cdot e^{x^2} , dx = \frac{1}{2} e^{x^2} + 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.
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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