Find the integral of (3x^2)(e^(x^3)) dx.

Find the integral of (3x^2)(e^(x^3)) dx.

• A. e^(x^3) + C
• B. (1/3)e^(x^3) + C
• C. 3e^(x^3) + C
• D. e^(x^3) - (1/3)e^(x^3) + C

Answer: (A)

To find the integral of \( (3x^2)(e^{x^3}) \, dx \), we can use integration by substitution. Let us substitute \( u = x^3 \). Then, the differential \( du = 3x^2 \, dx \). This means \( dx = \frac{du}{3x^2} \). We can express \( 3x^2 \, dx \) simply as \( du \). Rewriting the integral in terms of \( u \), we have: \[ \int (3x^2)(e^{x^3}) \, dx = \int e^{u} \, du \] The integral of \( e^{u} \) is simply \( e^{u} + C \), where \( C \) is the constant of integration. Now substituting back \( u = x^3 \): \[ e^{u} + C = e^{x^3} + C \] This leads us directly to the correct answer. Therefore, the correct choice is based on the fact that when integrating \( 3x^2 e^{x^3} \), we seamlessly transition through the substitution and arrive

To find the integral of ( (3x^2)(e^{x^3}) , dx ), we can use integration by substitution.

Let us substitute ( u = x^3 ). Then, the differential ( du = 3x^2 , dx ). This means ( dx = \frac{du}{3x^2} ). We can express ( 3x^2 , dx ) simply as ( du ).

Rewriting the integral in terms of ( u ), we have:

[

\int (3x^2)(e^{x^3}) , dx = \int e^{u} , du

]

The integral of ( e^{u} ) is simply ( e^{u} + C ), where ( C ) is the constant of integration. Now substituting back ( u = x^3 ):

[

e^{u} + C = e^{x^3} + C

]

This leads us directly to the correct answer.

Therefore, the correct choice is based on the fact that when integrating ( 3x^2 e^{x^3} ), we seamlessly transition through the substitution and arrive