How do you integrate #e^7x^3 x^2 dx#?
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To integrate (e^{7x^3} x^2 , dx), use the substitution method.
Let (u = 7x^3). Then, (du = 21x^2 , dx).
Notice that we almost have the right form, except we need (x^2 , dx), not (21x^2 , dx). To correct for this, divide both sides by 21: (du/21 = x^2 , dx).
Now, substitute back into the integral:
[ \int e^{7x^3} x^2 , dx = \int e^u \cdot \frac{1}{21} , du = \frac{1}{21} \int e^u , du ]
Integrating (e^u) with respect to (u) gives (e^u), so:
[ \frac{1}{21} \int e^u , du = \frac{1}{21} e^u + C ]
Substitute (u = 7x^3) back into the equation:
[ \frac{1}{21} e^{7x^3} + C ]
So, the integral of (e^{7x^3} x^2 , dx) is (\frac{1}{21} e^{7x^3} + 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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