# Find the integral $\int\frac{\large x^3+3x+4}{\large \sqrt x}\;dx$

$\begin{array}{1 1}\frac{2}{7} (x^\frac{7}{2})+2x^\frac{3}{2}+8x^\frac{1}{2}+c \\ \frac{2}{5} (x^\frac{5}{2})+2x^\frac{3}{2}+8x^\frac{1}{2}+c \\ \frac{2}{7} (x^\frac{7}{2})+2x^\frac{7}{2}+8x^\frac{1}{2}+c \\ \frac{2}{7} (x^\frac{7}{2}) + 2x^\frac{3}{2}+8x^\frac{7}{2}+c\end{array}$

## 1 Answer

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• $\int x^n dx=\frac{x^{n+1}}{n+1}+c\;$
$\int\frac{x^3+3x+4}{\sqrt x}dx.$
We can split this as,
$\;\;\;=\int\frac{x^3}{\sqrt x}dx+\int\frac{3x}{\sqrt x}dx+\int\large\frac{4}{\sqrt x}dx$$.$
$\;\;\;=\int x^{(3-\frac{1}{2})}dx+3\int x^{(1-\frac{1}{2}) }dx+4\int x^{-\frac{1}{2}}dx$.
$\;\;\;=\int x^\frac{5}{2}dx+3\int x^\frac{1}{2} dx+4\int x^\frac{-1}{2}dx.$
$\;\;\;=\begin{bmatrix}\frac{x^\frac{5}{2}}{\frac{5}{2}+1}\end{bmatrix}+\begin{bmatrix}\frac{x^\frac{1}{2}}{\frac{1}{2}+1}\end{bmatrix}+\begin{bmatrix}\frac{x^\frac{-1}{2}}{\frac{-1}{2}+1}\end{bmatrix}+c$
$\;\;\;=\begin{bmatrix}\frac{x^\frac{7}{2}}{\frac{7}{2}}\end{bmatrix}+3\begin{bmatrix}\frac{x^\frac{3}{2}}{\frac{3}{2}}\end{bmatrix}+4\begin{bmatrix}\frac{x^\frac{1}{2}}{\frac{1}{2}}\end{bmatrix}+c$
$\;\;\;=\frac{2}{7}(x^\frac{7}{2})+2x^\frac{3}{2}+8x^\frac{1}{2}+c$.
Hence $\int\frac{x^3+5x^2-4}{x^2}dx=\frac{2}{7}(x^\frac{7}{2})+2x^\frac{3}{2}+8x^\frac{1}{2}+c.$
answered Jan 27, 2013
edited Aug 9, 2013

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