What is the integral : `int x^3e^3x^2dx`

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justaguide eNotes educator| Certified Educator

The integral `int x^3*e^3*x^2 dx` has to be determined.

Let x^3 = y

=> `dy = 3*x^2 dx`

=> `(1/3)*dy = x^2* dx`

This changes the integral to `int (1/3)*e^3*y dy`

=> `(e^3/3)*(y^2)/2 + C`

substitute y = x^3

=> `(e^3/6)*x^6 + C`

The required integral is `(e^3/6)*x^6 + C`

beckden eNotes educator| Certified Educator

Since `e^3` is a constant we can write the integral as

`e^3 int x^5 dx`

Since i`int x^n dx = 1/(n+1) x^(n+1) + C`

Our answer is `e^3*(1/(5+1)x^(5+1)) + C = (e^3x^6)/6 + C`

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