The cosine of the angle included between the lines $$\mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$$ and $$\mathbf{r}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\mu(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})$$ where $$\lambda, \mu \in R$$ is.

$$\int \frac{1+2 e^{-x}}{1-2 e^{-x}} d x=$$

If the volume of the parallelopiped whose conterminus edges are along the vectors $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ is 12, then the volume of the tetrahedron whose conterminus edges are $$\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}$$ and $$c+a$$ is

The function $$f(x)=\frac{x+1}{9 x+x^3}$$ is