The given function is \(\displaystyle (\cos x+\sin x)^{2} \) and we need to find the integration of this function with respect to \( x \).
Solution:
\( \begin{align*} &\int (\cos x + \sin x)^2 \, dx \\ &= \int (\cos^2 x + 2 \sin x \cos x + \sin^2 x) \, dx \\ &= \int (1 + \sin 2x) \, dx \quad (\because \sin 2x = 2 \sin x \cos x) \\ &= \int 1 \, dx + \int \sin 2x \, dx \\ &= x \, – \frac{1}{2} \cos 2x + C, \quad \text{C is a constant} \\ \end{align*} \)Thus, the integration of the given function is
\( \boxed{\int (\cos x + \sin x)^2 \, dx = x \, – \frac{1}{2} \cos 2x + C} \\ \)Please let me know in the comments if you find any errors in this solution.