# 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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