Bonjour,
1/cos²(x) = sec(x)
[tex] \int\limits^{-\pi/4}_{\pi/4} \frac{tan(x)}{cos^2(x)} \, dx =\int\limits^{-\pi/4}_{\pi/4} tan(x)sec^2(x)dx[/tex]
Faisons le changement de variable suivant :
u = tan(x)
du = sec²(x)dx
[tex]\int\limits^{\pi/4}_{-\pi/4} \frac{tan(x)}{cos^2(x)} \, dx = \int\limits^1_{-1} u\times sec^2(x) \times \frac{1}{sec^2(x)}du \\\\
\int\limits^{\pi/4}_{-\pi/4} \frac{tan(x)}{cos^2(x)} \, dx = \int\limits^1_{-1} u \ du \\\\
\int\limits^{\pi/4}_{-\pi/4} \frac{tan(x)}{cos^2(x)} \, dx = \left[\frac{1}{2}u^2}\right]^1_{-1} \\\\
\int\limits^{\pi/4}_{-\pi/4} \frac{tan(x)}{cos^2(x)} \, dx =\frac{1}{2}\times 1^2-\frac{1}{2}\times (-1)^2\\\\
[/tex]
[tex]\boxed{\int\limits^{\pi/4}_{-\pi/4} \frac{tan(x)}{cos^2(x)} \, dx =0}[/tex]
