Question
Integrate the following functions w.r.t. x:
$\frac{3 x^3-2 x+5}{x \sqrt{x}}$
$\frac{3 x^3-2 x+5}{x \sqrt{x}}$
✓
Answer
$\int \frac{3 x^3-2 x+5}{x \sqrt{x}} d x$
$=\int x^{-\frac{3}{2}}\left(3 x^3-2 x+5\right) d x$
$=\int\left(3 x^{\frac{3}{2}}-2 x^{-\frac{1}{2}}+5 x^{-\frac{3}{2}}\right) d x$
$=3 \int x^{\frac{3}{2}} d x-2 \int x^{-\frac{1}{2}} d x+5 \int x^{-\frac{3}{2}} d x$
$=3\left(\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\right)-2\left(\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\right)+5\left(\frac{x^{-\frac{3}{2}+1}}{-\frac{3}{2}+1}\right)+c$
$=\frac{6}{5} x^2 \sqrt{x}-4 \sqrt{x}-\frac{10}{\sqrt{x}}+c$
$=\int x^{-\frac{3}{2}}\left(3 x^3-2 x+5\right) d x$
$=\int\left(3 x^{\frac{3}{2}}-2 x^{-\frac{1}{2}}+5 x^{-\frac{3}{2}}\right) d x$
$=3 \int x^{\frac{3}{2}} d x-2 \int x^{-\frac{1}{2}} d x+5 \int x^{-\frac{3}{2}} d x$
$=3\left(\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\right)-2\left(\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\right)+5\left(\frac{x^{-\frac{3}{2}+1}}{-\frac{3}{2}+1}\right)+c$
$=\frac{6}{5} x^2 \sqrt{x}-4 \sqrt{x}-\frac{10}{\sqrt{x}}+c$
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