Solve ∫ xf^n(x) wrt x

$$\int x f ^ { n } ( x ) d x$$

Evaluate

$$\frac{x^{3}f^{n}}{3}+С$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int x^{2}f^{n}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$f^{n}\int x^{2}\mathrm{d}x$$

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}.

$$f^{n}\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{f^{n}x^{3}}{3}$$

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.

$$\frac{f^{n}x^{3}}{3}+С$$

Differentiate w.r.t. x

$$x^{2}f^{n}$$