Solve ∫ (x^2+2x^2) wrt x

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

Evaluate

$$x^{3}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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}.

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

Simplify.

$$x^{3}$$

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.

$$x^{3}+С$$

Differentiate w.r.t. x

$$3x^{2}$$