What is x if integral sign (down 0 - up x) e^t(2e^t -3)dt=0?

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`int_0^xe^t(2e^t-3)dt=0`

`int_0^x(2e^(t+t)-3e^t)dt=0`

`int_0^x2e^(2t)dt-int_0^x3e^tdt=0`

`e^(2t)|_0^x-3e^t|_0^x=0`

`(e^(2x)-1)-3(e^x-1)=0`

`e^(2x)-3e^x+2=0`

Which is quadratic in e^x

e^(2x)-2e^x-e^x+2=0

(e^x-1)(e^x-2)=0

either

e^x-1=0

e^x=1

x=0

or

e^x-2=0

e^x=2

ln(2)=x

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