`int_0^(pi/4) (1 + cos^2(theta))/(cos^2(theta)) d theta` Evaluate the integral

Textbook Question

Chapter 5, 5.4 - Problem 37 - 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_0^(pi/4)(1+cos^2 theta)/(cos^2 theta) d theta =  int_0^(pi/4)1/(cos^2 theta) d theta + int_0^(pi/4) d theta`

`int_0^(pi/4)(1+cos^2 theta)/(cos^2 theta) d theta = (tan theta + theta)|_0^(pi/4)`

`int_0^(pi/4)(1+cos^2 theta)/(cos^2 theta) d theta = (tan (pi/4) + pi/4)-(tan 0 + 0)`

`int_0^(pi/4)(1+cos^2 theta)/(cos^2 theta) d theta = (1+ pi/4)`

Hence, evaluating the definite integral, yields `int_0^(pi/4)(1+cos^2 theta)/(cos^2 theta) d theta = (1+ pi/4).`

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