Solve ∫ (x^2+3x+e^x)

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

Evaluate

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

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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

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

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

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

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.

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

Differentiate w.r.t. x

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