g. use the substitution ( u = 5x + 4 ) to evaluate the integral ( int 10x + 11 \frac{dx}{sqrt{5x + 4}} ).

Explanation:

Step1: Find (du)

Given (u = 5x+4), then (du=5dx), so (dx=\frac{du}{5}). Also, (x=\frac{u - 4}{5}).

Step2: Substitute into the integral

[ \begin{align*} \int(10x + 11)\sqrt{5x + 4}dx&=\int\left(10\times\frac{u - 4}{5}+11\right)\sqrt{u}\times\frac{du}{5}\ &=\int\left(2(u - 4)+11\right)\sqrt{u}\times\frac{du}{5}\ &=\int(2u-8 + 11)\sqrt{u}\times\frac{du}{5}\ &=\int(2u + 3)u^{\frac{1}{2}}\times\frac{du}{5}\ &=\frac{1}{5}\int(2u^{\frac{3}{2}}+3u^{\frac{1}{2}})du \end{align*} ]

Step3: Integrate term - by - term

Using the power rule (\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have: [ \begin{align*} \frac{1}{5}\int(2u^{\frac{3}{2}}+3u^{\frac{1}{2}})du&=\frac{1}{5}\left(2\times\frac{u^{\frac{3}{2}+1}}{\frac{3}{2}+1}+3\times\frac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)+C\ &=\frac{1}{5}\left(\frac{4}{5}u^{\frac{5}{2}}+2u^{\frac{3}{2}}\right)+C\ &=\frac{4}{25}(5x + 4)^{\frac{5}{2}}+\frac{2}{5}(5x + 4)^{\frac{3}{2}}+C \end{align*} ]

Answer:

(\frac{4}{25}(5x + 4)^{\frac{5}{2}}+\frac{2}{5}(5x + 4)^{\frac{3}{2}}+C)