Solve ∫ (sinx+cosx+1) wrt x

$$\int ( \sin x + \cos x + 1 ) d x$$

Evaluate

$$\sin(x)-\cos(x)+x+С$$

Solution Steps

Integrate the sum term by term.

$$\int \sin(x)\mathrm{d}x+\int \cos(x)\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$-\cos(x)+\int \cos(x)\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$-\cos(x)+\sin(x)+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\cos(x)+\sin(x)+x$$

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.

$$-\cos(x)+\sin(x)+x+С$$

Differentiate w.r.t. x

$$\sin(x)+\cos(x)+1$$