What is the integral of (x^5-3x^2+4)/(sqrt(x)) ?

The integral of (x^5-3x^2+4)/(sqrt(x)) is 2/11 x^{11/2}-6/5 x^{5/2}+8sqrt(x)+C

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integral of (x^5-3x^2+4)/(sqrt(x))

\int \frac{x ^{5} - 3 x ^{2} + 4}{x} d x

\frac{2}{11} x^{\frac{11}{2}} - \frac{6}{5} x^{\frac{5}{2}} + 8 x + C

Solution steps

\int \frac{x ^{5} - 3 x ^{2} + 4}{x} d x

Expand

\frac{x ^{5} - 3 x ^{2} + 4}{x} : x^{\frac{9}{2}} - 3 x^{\frac{3}{2}} + \frac{4}{x}

= \int x^{\frac{9}{2}} - 3 x^{\frac{3}{2}} + \frac{4}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{\frac{9}{2}} d x - \int 3 x^{\frac{3}{2}} d x + \int \frac{4}{x} d x

\int x^{\frac{9}{2}} d x = \frac{2}{11} x^{\frac{11}{2}}

\int 3 x^{\frac{3}{2}} d x = \frac{6}{5} x^{\frac{5}{2}}

\int \frac{4}{x} d x = 8 x

= \frac{2}{11} x^{\frac{11}{2}} - \frac{6}{5} x^{\frac{5}{2}} + 8 x

Add a constant to the solution

= \frac{2}{11} x^{\frac{11}{2}} - \frac{6}{5} x^{\frac{5}{2}} + 8 x + C

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\int 11 d x

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\int sin (14 x) cos (11 x) d x

\int x^{1.1} + 7 x^{2.5} d x

What is the integral of (x^5-3x^2+4)/(sqrt(x)) ?
The integral of (x^5-3x^2+4)/(sqrt(x)) is 2/11 x^{11/2}-6/5 x^{5/2}+8sqrt(x)+C