Definite integralEvaluate the definite integral of y=1/cos ^2x. x=0 to x=pi/4

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should notice that you may directly integrate, using the following formula such that:

`int_0^(pi/4) 1/(cos^2 x)dx = tan x|_0^(pi/4)`

You need to use the fundamental theorem of calculus, such that:

`int_0^(pi/4) 1/(cos^2 x)dx = tan(pi/4) - tan 0`

`int_0^(pi/4) 1/(cos^2 x)dx = 1 - 0`

`int_0^(pi/4) 1/(cos^2 x)dx = 1`

Hence, evaluating the given definite integral, using formula of integration, yields `int_0^(pi/4) 1/(cos^2 x)dx = 1.`

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The definite integral will be evaluated using the Leibniz-Newton formula.

Int f(x)dx = F(b) - F(a), where x = a to x = b

We'll put y = f(x) = 1/(cos x)^2

We'll compute the indefinite integral, first:

Int dx/(cos x)^2 = tan x + C

We'll note the result F(x) = tan x + C

We'll determine F(a), for a = 0:

F(0) = tan 0

F(0) = 0

We'll determine F(b), for b = pi/4:

F(pi/4) = tan pi/4

F(pi/4) = 1

We'll evaluate the definite integral:

Int dx/(cos x)^2 = F(pi/4) - F(0)

Int dx/(cos x)^2 = 1 - 0

Int dx/(cos x)^2 = 1, from x = 0 to x = pi/4

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