How do you integrate #e^7x^3 x^2 dx#?

Answer 1

#=(e^7)/6 (x^6) +C #

#int e^7x^3x^2 \dx#
I can see two approaches 1) combining the #x# terms or 2) doing u substitution. first lets express this differently by moving out the constant
#e^7int x^3x^2 \dx# 1)
#e^7int x^5\\dx# #(e^7)/6 (x^6)+C#
2) now i will use u substitution and let #u=x^3# then #(du)/(dx) =3x^2# #1/3du=x^2 dx#
now substituting this back in #e^7int u1/3 \\du# #(e^7)/3int u \\du#
#=(e^7)/3 (u^2)/2 +C# now we place replace u #=(e^7)/6 (x^3)^2 +C # #=(e^7)/6 (x^6) +C #
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Answer 2

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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Answer from HIX Tutor

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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