Integral Secant C98E12

1. **State the problem:** We want to evaluate the definite integral $$\int_0^{\frac{\pi}{4}} \sec(t) \, dt.$$ 2. **Recall the formula:** The integral of $\sec(t)$ is $$\int \sec(t) \, dt = \ln|\sec(t) + \tan(t)| + C.$$ 3. **Apply the definite integral:** Using the formula, we have $$\int_0^{\frac{\pi}{4}} \sec(t) \, dt = \left[ \ln|\sec(t) + \tan(t)| \right]_0^{\frac{\pi}{4}}.$$ 4. **Evaluate at the bounds:** - At $t = \frac{\pi}{4}$, $$\sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2},$$ $$\tan\left(\frac{\pi}{4}\right) = 1.$$ So, $$\sec\left(\frac{\pi}{4}\right) + \tan\left(\frac{\pi}{4}\right) = \sqrt{2} + 1.$$ - At $t = 0$, $$\sec(0) = 1,$$ $$\tan(0) = 0,$$ so $$\sec(0) + \tan(0) = 1.$$ 5. **Calculate the difference:** $$\ln|\sqrt{2} + 1| - \ln|1| = \ln(\sqrt{2} + 1) - 0 = \ln(\sqrt{2} + 1).$$ **Final answer:** $$\int_0^{\frac{\pi}{4}} \sec(t) \, dt = \ln(\sqrt{2} + 1).$$