Solve ∫ [cos(x)+3sin(x)] wrt x

$$\int [ \cos ( x ) + 3 \sin ( x ) ] d x$$

Evaluate

$$\sin(x)-3\cos(x)+С$$

Solution Steps

Integrate the sum term by term.

$$\int \cos(x)\mathrm{d}x+\int 3\sin(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \cos(x)\mathrm{d}x+3\int \sin(x)\mathrm{d}x$$

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

$$\sin(x)+3\int \sin(x)\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result. Multiply 3 times -\cos(x).

$$\sin(x)-3\cos(x)$$

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.

$$\sin(x)-3\cos(x)+С$$

Differentiate w.r.t. x

$$\cos(x)+3\sin(x)$$