Q. 25 - Calculus

Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \frac{x+3}{2\sqrt{x}}\mathrm{d}x$

Step-by-Step Solution

Answer

The solution of the given integral is $\int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\frac{1}{3}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+3x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+C$.

Step 1. Given Information

Solving the given integrals.
$\int \frac{x+3}{2\sqrt{x}}\mathrm{d}x$

Step 2. Solving the given integral using algebra.

$$\begin{gathered} \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\int \left(\frac{x}{2\sqrt{x}}+\frac{3}{2\sqrt{x}}\right)\mathrm{d}x \\ \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\int \left(\frac{\sqrt{x}}{2}+\frac{3}{2x^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\right)\mathrm{d}x \\ \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\int \left(\frac{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2}+\frac{3x^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2}\right)\mathrm{d}x \end{gathered}$$

Step 3. After simplifying

$$\begin{gathered} \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\int \frac{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2}\mathrm{d}x+\int \frac{3x^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2}\mathrm{d}x \\ \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\frac{1}{2}\int x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{d}x+\frac{3}{2}\int x^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{d}x \\ \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\frac{1}{2}\left[\frac{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1}}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1}\right]+\frac{3}{2}\left[\frac{x^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1}}{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1}\right]+C \\ \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\frac{1}{2}\left[\frac{x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]+\frac{3}{2}\left[\frac{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]+C \\ \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\frac{1}{2}·\frac{2}{3}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\frac{3}{2}·\frac{2}{1}{x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+C \\ \int \frac{x+3}{2\sqrt{x}}\mathrm{d}x=\frac{1}{3}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+3x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+C \end{gathered}$$