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You may integrate directly, using the formula of integration, such that:
`int_0^(pi/4) 1/(sin^2 x)dx = -cot x|_0^(pi/4)`
You need to use fundamental theorem of calculus, such that:
`int_0^(pi/4) 1/(sin^2 x)dx = -cot(pi/4) - (-cot 0)`
`int_0^(pi/4) 1/(sin^2 x)dx = -1 + 1/0`
Since `cot 0` is not a valid value, hence, you cannot evaluate the definite integral `int_0^(pi/4) 1/(sin^2 x)dx` at having the limit of integration `x = 0` .
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 f(x) = 1/(sin x)^2
We'll compute the indefinite integral, first:
Int dx/(sin x)^2 = -cot x + C
We'll note the result F(x) = - cot x + C
We'll determine F(a), for a = 0:
F(0) = -cot 0
F(0) = -pi/2
We'll determine F(b), for b = pi/4:
F(pi/4) = -cot pi/4
F(pi/4) = -1
We'll evaluate the definite integral:
Int dx/(sin x)^2 = F(pi/4) - F(0)
Int dx/(sin x)^2 = -1+ pi/2
Int dx/(sin x)^2 = -1+ pi/2, from x = 0 to x = pi/4.
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