Solve ∫ (2x^2+e^x) wrt x

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

Evaluate

$$e^{x}+\frac{2x^{3}}{3}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2x^{2}\mathrm{d}x+\int e^{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{2}\mathrm{d}x+\int e^{x}\mathrm{d}x$$

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

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

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

$$\frac{2x^{3}}{3}+e^{x}$$

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.

$$\frac{2x^{3}}{3}+e^{x}+С$$

Differentiate w.r.t. x

$$2x^{2}+e^{x}$$