Fórmula:

$$\int f(x)g(x)dx=F(x)g(x)-\int G(x)g'(x)dx$$

1. $\int xe^xdx \\ \begin{array}{cccccccccc}\int &x&e^x&dx=&x&e^x&-\int&1&e^x&dx=xe^x-e^x+C\\&\uparrow&\uparrow&&\uparrow&\uparrow&&\uparrow&\uparrow&\\&f&g&&f&G&&f'&G&\\ \end{array}$
2. $\int x\sin x dx \\ \begin{array}{cccccccccc} \int &x&\sin x& dx=&x&(-\cos x)&-\int&1&(-\cos x)&dx=-x \cos x +\sin x +C\\&\uparrow&\uparrow&&\uparrow&\uparrow&&\uparrow&\uparrow&\\&f&g&&f&G&&f'&G&\\ \end{array}$
3. $\int \ln x dx \\ \begin{array}{cccccccccc}\int &1&\ln x& dx =&x& \ln x& - \int &x& \frac{1}{x}& dx = x \ln x -x + C\\ &\uparrow&\uparrow&&\uparrow&\uparrow&&\uparrow&\uparrow&\\&f&g&&f&G&&f'&G&\\ \end{array}$
4. $\int \frac{\ln x}{x}dx \\\int \frac{\ln x}{x}dx = \int \frac{1}{x}dx = \ln x \ln x - \int \ln x . \frac{1}{x}dx \\ \int \frac{\ln x}{x}dx =(\ln x)^2-\int \frac{\ln x}{x}dx \Leftrightarrow \int \frac{\ln x}{x}dx+\int \frac{\ln x}{x}dx=(\ln x)^2\Leftrightarrow \\ \Leftrightarrow \int \frac{\ln x}{x}dx=\frac{1}{2}(\ln x)^2+C$
5. $\int (\ln x)^2 dx\\ \int (\ln x)^2 dx=\int 1. (\ln x)^2 dx=x(\ln x)^2-\int x.2.\ln x . \frac{1}{x}=\\=x(\ln x)^2-2\int 1. \ln x dx = x(\ln x)^2 -2x \ln x +2 \int x. \frac {1}{x}dx =\\ = x(\ln x)^2 -2x\ln x + 2x+ C$