∫(π/4)^(5π/4) sec²(x) dx =

Explanation:

Step1: Recall antiderivative of $\sec^{2}(x)$

The antiderivative of $\sec^{2}(x)$ is $\tan(x)$.

Step2: Apply the fundamental theorem of calculus

$\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}}\sec^{2}(x)dx=\tan(x)\big|_{\frac{\pi}{4}}^{\frac{5\pi}{4}}$.

Step3: Evaluate the definite - integral

$\tan(\frac{5\pi}{4})-\tan(\frac{\pi}{4})$. Since $\tan(\frac{5\pi}{4}) = 1$ and $\tan(\frac{\pi}{4})=1$, we have $1 - 1=0$.

Answer:

$0$