Question

Evaluate the integral\(\)\(\int\limits_4^9 {\frac{{\ell ny}}{{\sqrt y }}} dy\)

Short answer

We will use theintegration by part method to solve this given integral

\(\int {\mu d\nu = \mu \nu } - \int {\nu .} d\mu \)

Step-by-step solution

Given Data

Given =\({\rm I} = \int\limits_4^9 {\frac{{\ell ny}}{{\sqrt y }}} dy\)

Let \(\mu = \frac{1}{{\sqrt y }}\)\(d\mu = \frac{{ - 1}}{{2{y^{\frac{3}{2}}}}}dy\)

\(\int {d\nu = \int {\ln ydy} } \) \(\nu = \frac{1}{y}\)

Use Integration by Part

\(\begin{aligned}{l}\int\limits_4^9 {\frac{{\ell ny}}{{\sqrt y }}} dy = \left( {\frac{1}{{y\sqrt y }}} \right)_4^9 - \int\limits_4^9 {\frac{1}{y}} \left( {\frac{{ - 1}}{{2{y^{\frac{3}{2}}}}}} \right)\\ = \left( {\frac{1}{{9\left( 3 \right)}} - \frac{1}{{4\left( 2 \right)}}} \right) + \frac{1}{2}\int\limits_4^9 {\frac{{dy}}{{{y^{\frac{5}{2}}}}}} \\ = \left( {\frac{1}{{27}} - \frac{1}{8}} \right) + \frac{1}{2}\left( {\frac{{ - 5}}{2}} \right)\left( {\frac{1}{{{y^{\frac{7}{2}}}}}} \right)_4^9\\ = \frac{{8 - 27}}{{216}} - \frac{5}{4}\left( {\frac{1}{{{3^7}}} - \frac{1}{{{2^7}}}} \right)\\ = \frac{{ - 19}}{{216}} - \left( { - 0.009} \right)\\ = - 0.088 + 0.009\\ = - 0.079\end{aligned}\)

Hence, \(\int\limits_4^9 {\frac{{\ell ny}}{{\sqrt y }}} dy = - 0.079\)