Solve ∫ e^x+3x

$$\int e ^ { x } + 3 x$$

Evaluate

$$e^{x}+\frac{3x^{2}}{2}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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}.

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

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.

$$e^{x}+\frac{3x^{2}}{2}+С$$

Differentiate w.r.t. x

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