`int_0^1(x^e + e^x)dx` Evaluate the integral.

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Chapter 5, 5.3 - Problem 37 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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lemjay | High School Teacher | (Level 3) Senior Educator

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`int_0^1 (x^e+e^x)dx`

To evaluate this, apply the formulas

`int u^ndu = u^(n+1)/(n+1)`   and   `int e^udu=e^u` .

`= (x^(e+1)/(e+1) + e^x) |_0^1`

Then, plug-in the limits of integral as follows `F(x) = int_a^b f(x) dx = F(b)-F(a)` .

`= (1^(e+1)/(e+1) + e^1)- (0^(e+1)/(e+1) + e^0)`

`= (1/(e+1) + e) - (0+1)`

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

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

`=1/(e+1)+(e^2+e)(e+1) -(e+1)/(e+1)`

`=(1+e^2+e-e-1)/(e+1)`

`=e^2/(e+1)`

Therefore, `int _0^1 (x^e+e^x)dx = e^2/(e+1)` .

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