Solve ∫ (3y^2+e^y) wrt y

$$\int ( 3 y ^ { 2 } + e ^ { y } ) d y$$

Evaluate

$$y^{3}+e^{y}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 3y^{2}\mathrm{d}y+\int e^{y}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$3\int y^{2}\mathrm{d}y+\int e^{y}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}. Multiply 3 times \frac{y^{3}}{3}.

$$y^{3}+\int e^{y}\mathrm{d}y$$

Use \int e^{y}\mathrm{d}y=e^{y} from the table of common integrals to obtain the result.

$$y^{3}+e^{y}$$

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.

$$y^{3}+e^{y}+С$$

Differentiate w.r.t. y

$$3y^{2}+e^{y}$$