Solve any maths problem

Step by step

1. \int x \log{\left(x \right)}\, dx

   Find an antiderivative.

2. u = \log{\left(x \right)},\quad du = \frac{1}{x}\, dx

   Substitute u = \log{\left(x \right)}.

3. \int x \log{\left(x \right)}\, dx = \int u e^{2 u}\, d_u

   Rewrite the integral in terms of u.

4. u = u,\quad dv = e^{2 u}\, d_u

   Integration by parts: ∫u dv = uv − ∫v du.

5. du = 1\, d_u,\quad v = \frac{e^{2 u}}{2}

   Differentiate u, integrate dv.

6. \int u e^{2 u}\, d_u = \frac{u e^{2 u}}{2} - \int \frac{e^{2 u}}{2}\, d_u

   Apply the formula.

7. \int \frac{e^{2 u}}{2}\, d_u = \frac{1}{2} \int e^{2 u}\, d_u

   Pull the constant \frac{1}{2} out of the integral.

8. u = 2 u,\quad du = 2\, d_u

   Substitute u = 2 u.

9. \int e^{2 u}\, d_u = \int \frac{e^{u}}{2}\, d_u

   Rewrite the integral in terms of u.

10. \int \frac{e^{u}}{2}\, d_u = \frac{1}{2} \int e^{u}\, d_u

    Pull the constant \frac{1}{2} out of the integral.

11. \int e^{u}\, d_u = e^{u}

    ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).

12. = \frac{e^{2 u}}{2}

    Substitute back u = 2 u.

13. = \frac{u e^{2 u}}{2} - \frac{e^{2 u}}{4}

    Combine.

14. = \frac{x^{2} \log{\left(x \right)}}{2} - \frac{x^{2}}{4}

    Substitute back u = \log{\left(x \right)}.

15. F(x) = \frac{x^{2} \left(2 \log{\left(x \right)} - 1\right)}{4} + C

    Add the constant of integration.