`int_(1/sqrt(3))^sqrt(3)(8/(1 + x^2))dx` Evaluate the integral.

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Chapter 5, 5.3 - Problem 39 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the definite integral using the fundamental theorem of calculus, such that:

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

`int_(1/(sqrt3))^(sqrt 3) 8/(1+x^2) dx = 8 int_(1/(sqrt3))^(sqrt 3) 1/(1+x^2) dx`

Using the formula `int 1/(a^2 + x^2) dx = (x/a) arctan (x/a) + c` yields:

`8 int_(1/(sqrt3))^(sqrt 3) 1/(1+x^2) dx = 8arctan x|_(1/(sqrt3))^(sqrt 3)`

`8 int_(1/(sqrt3))^(sqrt 3) 1/(1+x^2) dx = 8(arctan sqrt 3 - arctan (1/(sqrt3)))`

`8 int_(1/(sqrt3))^(sqrt 3) 1/(1+x^2) dx = 8(pi/3 - pi/6) = 8pi/6 = 4pi/3`

Hence, evaluating the definite integral, yields `8 int_(1/(sqrt3))^(sqrt 3) 1/(1+x^2) dx = 4pi/3.`

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