Solve ∫ x^2xuf(u)du

$$\int x ^ { 2 x } u f ( u ) d u$$

Evaluate

$$\frac{fu^{3}x^{2x}}{3}+С$$

Solution Steps

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

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

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

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

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

$$x^{2x}f\times \frac{u^{3}}{3}$$

Simplify.

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

If F\left(u\right) is an antiderivative of f\left(u\right), then the set of all antiderivatives of f\left(u\right) is given by F\left(u\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{x^{2x}fu^{3}}{3}+С$$

Differentiate w.r.t. x

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