Solve ∫ (5cosx-4sinx) wrt x

$$\int ( 5 \cos x - 4 \sin x ) d x$$

Evaluate

$$4\cos(x)+5\sin(x)+С$$

Solution Steps

Integrate the sum term by term.

$$\int 5\cos(x)\mathrm{d}x+\int -4\sin(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int \cos(x)\mathrm{d}x-4\int \sin(x)\mathrm{d}x$$

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

$$5\sin(x)-4\int \sin(x)\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply -4 times -\cos(x).

$$5\sin(x)+4\cos(x)$$

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.

$$5\sin(x)+4\cos(x)+С$$

Differentiate w.r.t. x

$$5\cos(x)-4\sin(x)$$