Answer
$a.\quad \displaystyle \frac{506}{375}$
$b.\quad\displaystyle \frac{86,744}{375}$
Work Step by Step
$a.$
Applying th.7, let $u=g(t)=4+5t$ (a continuous function).
$(t=\displaystyle \frac{u-4}{5})$
Then, $\displaystyle \quad du=g'(t)dt=5dt,\qquad dt= \frac{du}{5} $
and, the borders change to $g(0)=4$ and $g(1)=9$.
$\displaystyle \int_{0}^{1}t\sqrt{4+5t}dt=\int_{4}^{9} \frac{u-4}{5}\cdot\sqrt{u}\cdot\frac{du}{5} $
$=\displaystyle \frac{1}{25}\int_{4}^{9}\left[u\sqrt{u}-4 \sqrt{u} \right] du$
$=\displaystyle \frac{1}{25}\int_{4}^{9}\left[u^{3/2}-4 u^{-1/2} \right] du$
$=\displaystyle \frac{1}{25}\left[ \frac{u^{3/2+1}}{3/2+1}-4\cdot\frac{u^{1/2+1}}{1/2+1} \right]_{4}^{9}$
$=\displaystyle \frac{1}{25}\left[ \frac{2u^{5/2}}{5}-\frac{8u^{3/2}}{3} \right]_{4}^{9}$
$=\displaystyle \frac{1}{25}\left[ (\frac{2\cdot 9^{5/2}}{5}-\frac{8\cdot 9^{3/2}}{3})-(\frac{2\cdot 4^{5/2}}{5}-\frac{8\cdot 4^{3/2}}{3}) \right]$
$=\displaystyle \frac{1}{25}\left[ (\frac{2\cdot 3^{5}}{5}-\frac{8\cdot 3^{3}}{3})-(\frac{2\cdot 2^{5}}{5}-\frac{8\cdot 2^{3}}{3}) \right]$
$=\displaystyle \frac{1}{25}\left[ (\frac{2\cdot 243}{5}-\frac{8\cdot 27}{3})-(\frac{2\cdot 32}{5}-\frac{8\cdot 8}{3}) \right]$
$=\displaystyle \frac{1}{25}\cdot (\frac{2\cdot 243\cdot 3-8\cdot 27\cdot 5-2\cdot 32\cdot 3+8\cdot 8\cdot 5}{15}) $
$=\displaystyle \frac{506}{375}$
$b.$
Using the same substitution, $u=4+5t,\ du=5dt,$
the borders here change to
$g(1)=9$ and $g(49)=9$.
$\displaystyle \int_{0}^{1}t\sqrt{4+5t}dt=\int_{9}^{49} \frac{u-4}{5}\cdot\sqrt{u}\cdot\frac{du}{5} $
Following the same steps as above
$=\displaystyle \frac{1}{25}\left[ \frac{2u^{5/2}}{5}-\frac{8u^{3/2}}{3} \right]_{9}^{49}$
$=\displaystyle \frac{1}{25}\left[ (\frac{2\cdot 49^{5/2}}{5}-\frac{8\cdot 49^{3/2}}{3})-(\frac{2\cdot 9^{5/2}}{5}-\frac{8\cdot 9^{3/2}}{3}) \right]$
$=\displaystyle \frac{1}{25}\left[ (\frac{2\cdot 7^{5}}{5}-\frac{8\cdot 7^{3}}{3})-(\frac{2\cdot 3^{5}}{5}-\frac{8\cdot 3^{3}}{3}) \right]$
$=\displaystyle \frac{1}{25}\cdot\frac{2\cdot 7^{5}\cdot 3-8\cdot 7^{3}\cdot 5-2\cdot 3^{5}\cdot 3+8\cdot 3^{3}\cdot 5}{15}$
$=\displaystyle \frac{86,744}{375}$
