Solve ∫ x^3(x+1)(x+2) wrt x

$$\int{ { x }^{ 3 } (x+1)(x+2) }d x$$

Evaluate

$$\frac{x^{6}}{6}+\frac{3x^{5}}{5}+\frac{x^{4}}{2}+С$$

Solution Steps

Use the distributive property to multiply x^{3} by x+1.

$$\int \left(x^{4}+x^{3}\right)\left(x+2\right)\mathrm{d}x$$

Use the distributive property to multiply x^{4}+x^{3} by x+2 and combine like terms.

$$\int x^{5}+3x^{4}+2x^{3}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{5}\mathrm{d}x+\int 3x^{4}\mathrm{d}x+\int 2x^{3}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{5}\mathrm{d}x+3\int x^{4}\mathrm{d}x+2\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}.

$$\frac{x^{6}}{6}+3\int x^{4}\mathrm{d}x+2\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply 3 times \frac{x^{5}}{5}.

$$\frac{x^{6}}{6}+\frac{3x^{5}}{5}+2\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 2 times \frac{x^{4}}{4}.

$$\frac{x^{6}}{6}+\frac{3x^{5}}{5}+\frac{x^{4}}{2}$$

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.

$$\frac{x^{6}}{6}+\frac{3x^{5}}{5}+\frac{x^{4}}{2}+С$$

Differentiate w.r.t. x

$$\left(x+1\right)\left(x+2\right)x^{3}$$