The area of the region bounded by the curve $$y=4 x^3-6 x^2+4 x+1$$ and the lines $$x=1, x=5$$ and $$X$$-axis is

$$\int_\limits2^3 \frac{x}{x^2-1} d x=$$

The angles between the lines $$\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \text { and } \mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda^{\prime}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), \lambda, \lambda^{\prime} \in \mathbf{R}$$ is

The p.d.f of c.r.v $$X$$ is given by $$f(x)=\frac{x+2}{18}$$, if $$-2