Solve ∫ x(x+1)^3 wrt x=?

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

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

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

Use the distributive property to multiply x by x^{3}+3x^{2}+3x+1.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

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

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

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

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}.

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

Simplify.

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

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.

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

Differentiate w.r.t. x

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