Fundamentally different

$F(x) = \displaystyle \int_{0}^{x} \sec^2t dt = \tan t |_{0}^{x} = \tan x$

$G(x) = \displaystyle \int_{0}^{x} \tan t dt = \ln(\sec t) |_{0}^{x} = \ln(\sec x)$

$I = \displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \ln(\sec x) dx = 2 \int_{0}^{\frac{\pi}{2}} \ln(\sec x) dx = -2 \int_{0}^{\frac{\pi}{2}} \ln(\cos x) dx$

I consider this as a property :

$\displaystyle \int_{0}^{\frac{\pi}{2}} \ln(\cos x) dx = \dfrac{-\pi}{2} \ln 2$

Substituting the above property in the previous step involving finding $I$ ,

$I = -2 \times \dfrac{-\pi}{2} \ln 2 = \pi \ln 2 \approx \boxed{2.178}$

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Note:

• Since $\ln(\sec x)$ is an even function, I used the property of an even function, say $f(x)$ , which is:

$\displaystyle \int_{-a}^{a} f(x) dx = 2\int_{0}^{a} f(x) dx$

in the third line.