# Let

Question:

Let $I_{n}=\int_{1}^{e} x^{19}(\log |x|)^{n} d x$, where $n \in N$. If (20) $\mathrm{I}_{10}=\alpha \mathrm{I}_{9}+\beta \mathrm{I}_{8}$, for natural numbers $\alpha$ and $\beta$, then $\alpha-\beta$ equal to_________.

Solution:

Let $\overrightarrow{\mathrm{x}}=\lambda \overrightarrow{\mathrm{a}}+\mu \overrightarrow{\mathrm{b}} \quad(\lambda$ and $\mu$ are scalars $)$

$\overrightarrow{\mathrm{x}}=\hat{\mathrm{i}}(2 \lambda+\mu)+\hat{\mathrm{j}}(2 \mu-\lambda)+\hat{\mathrm{k}}(\lambda-\mu)$

Since $\overrightarrow{\mathrm{x}} \cdot(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})=0$

$3 \lambda+8 \mu=0$

Also Projection of $\vec{x}$ on $\vec{a}$ is $\frac{17 \sqrt{6}}{2}$

$\frac{\overrightarrow{\mathrm{x}} \cdot \overrightarrow{\mathrm{a}}}{|\overrightarrow{\mathrm{a}}|}=\frac{17 \sqrt{6}}{2}$

$6 \lambda-\mu=51$

$\lambda=8, \mu=-3$

From (1) and (2) $\quad \overrightarrow{\mathrm{x}}=13 \hat{\mathrm{i}}-14 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}$

$|\overrightarrow{\mathrm{x}}|^{2}=486$