Solve ∫ (cosy+y) wrt y

$$\int ( \cos y + y ) d y$$

Evaluate

$$\sin(y)+\frac{y^{2}}{2}+С$$

Solution Steps

Integrate the sum term by term.

$$\int \cos(y)\mathrm{d}y+\int y\mathrm{d}y$$

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

$$\sin(y)+\int y\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y\mathrm{d}y with \frac{y^{2}}{2}.

$$\sin(y)+\frac{y^{2}}{2}$$

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

$$\sin(y)+\frac{y^{2}}{2}+С$$

Differentiate w.r.t. y

$$\cos(y)+y$$