Single Variable Calculus, Chapter 5, 5.4, Section 5.4, Problem 32

Find the integrals $\displaystyle \int^9_1 \frac{3x-2}{\sqrt{x}} dx$

$\begin{equation} \begin{aligned} \int\frac{3x-2}{\sqrt{x}} dx &= \int \left( \frac{3x}{\sqrt{x}} - \frac{2}{\sqrt{x}} \right) dx \\ \\ \int\frac{3x-2}{\sqrt{x}} dx &= \int \left( 3x^{\frac{1}{2}} - 2x^{\frac{-1}{2}} \right) dx \\ \\ \int\frac{3x-2}{\sqrt{x}} dx &= 3 \int x^{\frac{1}{2}} dx - 2 \int x^{\frac{-1}{2}} dx\\ \\ \int\frac{3x-2}{\sqrt{x}} dx &= 3 \left( \frac{x^{\frac{1}{2} +1 }}{\frac{1}{2} +1 } \right) - 2 \left( \frac{x^{\frac{-1}{2}+1}}{\frac{-1}{2}+1 } \right) + C\\ \\ \int\frac{3x-2}{\sqrt{x}} dx &= 2 x^{\frac{3}{2}} - 2 \left( 2x^{\frac{1}{2}} \right) + C\\ \\ \int\frac{3x-2}{\sqrt{x}} dx &= 2x^{\frac{3}{2}} - 4x^{\frac{1}{2}} + C\\ \\ \int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 2(9)^{\frac{3}{2}} - 4 (9)^{\frac{1}{2}} + C- \left[ 2(1)^{\frac{3}{2}} - 4(1)^{\frac{1}{2}} + C \right]\\ \\ \int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 2\left[(9)^{\frac{1}{2}} \right]^3 - 4 (3) +C - 2 + 4 -C \\ \\ \int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 2(3)^3 - 12 + 2\\ \\ \int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 54 - 12 + 2\\ \\ \int^9_1 \frac{3x-2}{\sqrt{x}} dx &= 44 \end{aligned} \end{equation}$