Solve ∫ -sinx+1/2cosx

$$\int - \sin x + 1 / 2 \cos x$$

Evaluate

$$\frac{\sin(x)}{2}+\cos(x)+С$$

Solution Steps

Integrate the sum term by term.

$$\int -\sin(x)\mathrm{d}x+\int \frac{\cos(x)}{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\int \sin(x)\mathrm{d}x+\frac{\int \cos(x)\mathrm{d}x}{2}$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply -1 times -\cos(x).

$$\cos(x)+\frac{\int \cos(x)\mathrm{d}x}{2}$$

Use \int \cos(x)\mathrm{d}x=\sin(x) from the table of common integrals to obtain the result.

$$\cos(x)+\frac{\sin(x)}{2}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\cos(x)+\frac{\sin(x)}{2}+С$$

Differentiate w.r.t. x

$$\frac{\cos(x)-2\sin(x)}{2}$$