To integrate the function cos(1)x dx, we first recognize that the expression contains a constant multiplied by the variable x. The 1 is simply a constant, and we can express the integral as follows:
∫ cos(1)x dx = ∫ cos(kx) dx, where k = 1.
The integral of cos(kx) with respect to x is given by the formula:
∫ cos(kx) dx = (1/k) * sin(kx) + C, where C is the constant of integration.
For our case, since k = 1:
∫ cos(1)x dx = (1/1) * sin(1)x + C = sin(1)x + C.
Thus, the result of the integral is:
sin(1)x + C
In conclusion, integrating cos(1)x dx results in sin(1)x + C. This shows that we calculated the integral by applying the integration rules for trigonometric functions correctly, bearing in mind the constant factor.