Solve ∫ (sinx+7cosx-1) wrt x

$$\int ( \sin x + 7 \cos x - 1 ) d x$$

Evaluate

$$7\sin(x)-\cos(x)-x+С$$

Solution Steps

Integrate the sum term by term.

$$\int \sin(x)\mathrm{d}x+\int 7\cos(x)\mathrm{d}x+\int -1\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \sin(x)\mathrm{d}x+7\int \cos(x)\mathrm{d}x+\int -1\mathrm{d}x$$

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

$$-\cos(x)+7\int \cos(x)\mathrm{d}x+\int -1\mathrm{d}x$$

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

$$-\cos(x)+7\sin(x)+\int -1\mathrm{d}x$$

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

$$-\cos(x)+7\sin(x)-x$$

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.

$$-\cos(x)+7\sin(x)-x+С$$

Differentiate w.r.t. x

$$\sin(x)+7\cos(x)-1$$