How is calculate definite integral in y=e^x/(e^2x+1) between 0 and 0.5?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should come up with the following substitution such that: `e^x = t =gt e^x dx = dt` .

If you change the variable, you should calculate the limits of integration.

`x = 0 =gt e^0 = 1 = t`

`` `x = 0.5 =gt sqrt e = t`

Writing the integral yields:

`int_0^(0.5) (e^xdx)/(e^(2x) + 1) = int_1^(sqrte)(dt)/(t^2 + 1)`

`` `int_1^(sqrte) (dt)/(t^2 + 1) = arctant |_1^(sqrte)`

`int_1^(sqrte) (dt)/(t^2 + 1) = arctan sqrt e - arctan 1`

`int_0^(0.5) (e^xdx)/(e^(2x) + 1) = arctan sqrt e - pi/4`

Hence, evaluating the definite integral yields `int_0^(0.5) (e^xdx)/(e^(2x) + 1) = arctan sqrt e - pi/4.`

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