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What is #f(x) = int 3x^3-2x+xe^x dx# if #f(1) = 3 #?

Answer 1

Christopher Allers

The answer is #=3/4x^4-2/2x^2+e^x(x-1)+13/4#

We need

#intx^ndx=x^(n+1)/(n+1)+C(x!=-1)#

We start, by calculating

#intxe^xdx#

by integration by parts

#intuv'dx=uv-intu'vdx#

#u=x#, #=>#, #u'=1#

#v'=e^x#, #=>#, #v=e^x#

Therefore,

#intxe^xdx=xe^x-inte^xdx=xe^x-e^x#

So,

#f(x)=int(3x^3-2x+xe^x)dx#

#=3intx^3dx-2intxdx+intxe^xdx#

#=3/4x^4-2/2x^2+e^x(x-1)+C#

#f(1)=3/4-1+0+C=3#

#C=3+1/4=13/4#

Therefore,

#f(x)=3/4x^4-2/2x^2+e^x(x-1)+13/4#

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Answer 2

Isabella Ackerman

To find ( f(x) = \int (3x^3 - 2x + xe^x) , dx ) given ( f(1) = 3 ):

First, differentiate ( f(x) ) to find the original function ( f(x) ).

( f'(x) = 3x^3 - 2x + xe^x )

Then, integrate ( f'(x) ) to find ( f(x) ).

( f(x) = \int (3x^3 - 2x + xe^x) , dx = x^3 - x^2 + xe^x - e^x + C )

Using the given condition ( f(1) = 3 ):

( 3 = (1)^3 - (1)^2 + (1)e^1 - e^1 + C )

( 3 = 1 - 1 + e - e + C )

( 3 = 0 + C )

( C = 3 )

Therefore, the function ( f(x) ) is:

( f(x) = x^3 - x^2 + xe^x - e^x + 3 )

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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.