Integrate sin(4x+π/6) dx
Using the substitution u = 4x + π/6, du/dx = 4, dx = du/4. Therefore,
∫sin(4x+π/6) dx = ∫sin(u) (1/4) du
Integrating sin(u) gives -cos(u) + C, where C is the constant of integration. Substituting u back in gives:
- (1/4)cos(4x + π/6) + C
So the final answer is:
∫sin(4x+π/6) dx = -(1/4)cos(4x + π/6) + C
To integrate sin(4x + π/6) with respect to x, you can use the u-substitution method.
Let's assign a variable u, such that u = 4x + π/6.
First, differentiate both sides with respect to x to find du/dx:
du/dx = 4
Now, solve for dx by rearranging the equation:
dx = du/4
Substitute u and dx back into the integral:
∫ sin(4x + π/6) dx = ∫ sin(u) (du/4)
Since the derivative of sin(x) is -cos(x), the integral becomes:
∫ sin(u) (du/4) = -(1/4) ∫ cos(u) du
Now, integrate the function with respect to u:
-(1/4) ∫ cos(u) du = -(1/4) sin(u) + C
Don't forget to add the constant of integration, denoted by C.
Replace u with 4x + π/6 again:
-(1/4) sin(4x + π/6) + C
Therefore, the indefinite integral of sin(4x + π/6) dx is -(1/4) sin(4x + π/6) + C, where C is a constant.