# How do you find the definite integral for: #(x^43) (e^(-x^(44)) dx)# for the intervals #[0, 1]#?

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To find the definite integral of ( x^{43} e^{-x^{44}} ) over the interval [0, 1], you can use integration by substitution. Let ( u = -x^{44} ). Then, ( du = -44x^{43} dx ). Rearranging gives ( dx = -\frac{1}{44x^{43}} du ). Substitute these expressions into the integral:

[ \int_{0}^{1} x^{43} e^{-x^{44}} dx = -\int_{0}^{-1} e^u du ]

This simplifies the integral to ( -\left[ e^u \right]_{0}^{-1} ), which becomes ( -(e^{-1} - e^0) ). Thus, the definite integral is ( e - 1 ).

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