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Calculate antiderivative of y=3x*(e raised to 3x)

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rouche | Student, Undergraduate | (Level 1) Honors

Posted December 4, 2011 at 12:32 AM via web

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Calculate antiderivative of y=3x*(e raised to 3x)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 4, 2011 at 12:46 AM (Answer #1)

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You should evaluate the integral of the function `y=3x*e^(3x).` Since you detect that the function y is a product of two simpler functions, `3x`  and `e^(3x), ` you should use integration by parts.

Remember the formula of integration by parts:

u*v = int udv + int vdu

Put `u = 3x =gt du = 3dx`

Put `dv = e^(3x) dx=gt v = (e^(3x))/3`

Therefore you will have:

`3x*(e^(3x))/3 = int 3x*(e^(3x))dx + int (3*(e^(3x))/3) dx`

Separate the integral you should evaluate from the rest of terms.

`int 3x*(e^(3x))dx = 3x*(e^(3x))/3 - int (e^(3x)) dx`

`int 3x*(e^(3x))dx = 3x*(e^(3x))/3 - (e^(3x))/3 + c`

`int 3x*(e^(3x))dx = (e^(3x))/3* (3x-1) + c`

The antiderivative of the function y: `int 3x*(e^(3x))dx = (e^(3x))/3* (3x-1) + c`

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