The value of the integral ∫0nπ+t (|cos⁡ x| + |sin x|) dx is

Since the period of |sin⁡ x|+|cos⁡ x is π/2∫0nπ+t (|sin⁡ x|+|cos⁡ x|)dx=2n∫0π/2 (|sin⁡ x|+|cos⁡ x|)dx+∫0t (|sin⁡ x|+|cos⁡ x|)dx=2n∫0π/2 (sin⁡ x+cos⁡ x)dx+∫0t (sin⁡ x+cos⁡ x)dx=(2n)(2)+sin⁡ t−cos⁡ t+1=(4n+1)+sin⁡ t−cos⁡ t.