Find  `int_(-1)^1 -e^x dx`  using the Fundamental Theorem of Calculus. 1. S(superscript(1), subscript(-1)) -e^x dx and 2. S(superscript(4), subscript(1) sqrt(2/x) dx Thanks so much!!  

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To evaluate a definite integral, we need to use the Fundamental Theorem of Calculus. The formula is:

`int_a^b f(x) dx = F(x) | _a^b = F(b) - F(a)` 

where F(x) is the anti-derivative of f(x).

So, determine the the antiderivative of `-e^x` . Apply the indefinite integral formula `int e^u du = e^u + C` .

`int -e^x dx= - int e^x dx = -e^x + C`

Hence, the antiderivative of `-e^x` is `-e^x` ` ` . Then, substitute this to the formula of Fundamental Theorem of Calculus to evaluate the given definite integral.


`int_(-1)^1 -e^x = -e^x | _(-1)^1 = -e^1 - (-e^(-1) )`


            `= -e + e^(-1) = e^(-1) - e = 1/e - e`

Thus, `int_(-1)^1 -e^x = 1/e - e` .

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