I'm sorry, but I simply cannot figure out what that integral is intended to be.
Is that [tex]\int_{-1}^0 e^{\sqrt{x+1}} dx[/tex] or [tex]\int_{-1}^0 e^{\sqrt{x}+ 1} dx[/tex] or [tex]\int_{-1}^0 e^{\sqrt{x}}+ 1 dx[/tex]?
If it is the first, take [itex]u= \sqrt{x+1}= (x+1)^{1/2}[/itex] so that [itex]du= (1/2)(x+1)^{-1/2}dx[/itex] so that [itex](x+1)^{1/2}du= u du= dx[/itex]. When x= -1, u= 0, when x= 0, u= 1 The integral becomes
[tex]\int_0^1 ue^{u}du[/tex]
and can be done by "integration by parts".
If the second, write it as [tex]e\int_{-1}^0 e^{\sqrt{x}}dx[/tex] and let [itex]u= \sqrt{x}[/itex].
If the third, write it as [tex]\int_{-1}^0 e^{\sqrt{x}}dx+ \int_{-1}^0 dx[/tex].
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