26. \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} \cos \left(2 t+\frac{\pi}{4}\right) d t \)

To solve the integral \[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} \cos \left(2 t+\frac{\pi}{4}\right) d t, \] we start by using the property of cosine, which states that \(\cos(A + B) = \cos A \cos B - \sin A \sin B\). Here, we can set \(A = 2t\) and \(B = \frac{\pi}{4}\). Using this identity, we can rewrite the cosine function: \[ \cos \left(2 t + \frac{\pi}{4}\right) = \cos(2t)\cos\left(\frac{\pi}{4}\right) - \sin(2t)\sin\left(\frac{\pi}{4}\right). \] Since \(\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), we substitute this into our equation: \[ \cos\left(2 t + \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \cos(2t) - \frac{\sqrt{2}}{2} \sin(2t). \] Thus, the integral becomes: \[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} \left(\frac{\sqrt{2}}{2} \cos(2t) - \frac{\sqrt{2}}{2} \sin(2t)\right) dt. \] We can factor out \(\frac{\sqrt{2}}{2}\): \[ \frac{\sqrt{2}}{2} \left(\int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} \cos(2t) dt - \int_{-\frac{\pi}{4}}^{\frac{\pi}{2}} \sin(2t) dt\right). \] Now, we need to evaluate the two integrals separately. 1. **Integral of \(\cos(2t)\)**: The integral of \(\cos(2t)\) is \(\frac{1}{2} \sin(2t)\): \[ \int \cos(2t) dt = \frac{1}{2} \sin(2t). \] Evaluating it from \(-\frac{\pi}{4}\) to \(\frac{\pi}{2}\): \[ \left. \frac{1}{2} \sin(2t) \right|_{-\frac{\pi}{4}}^{\frac{\pi}{2}} = \frac{1}{2} \left(\sin(\pi) - \sin\left(-\frac{\pi}{2}\right)\right) = \frac{1}{2} (0 - (-1)) = \frac{1}{2}. \] 2. **Integral of \(\sin(2t)\)**: The integral of \(\sin(2t)\) is \(-\frac{1}{2} \cos(2t)\): \[ \int \sin(2t) dt = -\frac{1}{2} \cos(2t). \] Evaluating from \(-\frac{\pi}{4}\) to \(\frac{\pi}{2}\): \[ \left. -\frac{1}{2} \cos(2t) \right|_{-\frac{\pi}{4}}^{\frac{\pi}{2}} = -\frac{1}{2} \left(\cos(\pi) - \cos\left(-\frac{\pi}{2}\right)\right) = -\frac{1}{2} (-1 - 0) = \frac{1}{2}. \] Putting it all together, we substitute back into our original expression: \[ \frac{\sqrt{2}}{2} \left(\frac{1}{2} - \frac{1}{2}\right) = \frac{\sqrt{2}}{2} \cdot 0 = 0. \] Thus, the value of the integral is \[ \boxed{0}. \]