What is the integral of (4+6x-3x^2)/(sqrt(x)) ?

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

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integral of (4+6x-3x^2)/(sqrt(x))

\int \frac{4 + 6 x - 3 x ^{2}}{x} d x

8 x + 4 x^{\frac{3}{2}} - \frac{6}{5} x^{\frac{5}{2}} + C

Solution steps

\int \frac{4 + 6 x - 3 x ^{2}}{x} d x

Expand

\frac{4 + 6 x - 3 x ^{2}}{x} : \frac{4}{x} + 6 x - 3 x^{\frac{3}{2}}

= \int \frac{4}{x} + 6 x - 3 x^{\frac{3}{2}} 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 \frac{4}{x} d x + \int 6 x d x - \int 3 x^{\frac{3}{2}} d x

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

\int 6 x d x = 4 x^{\frac{3}{2}}

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

= 8 x + 4 x^{\frac{3}{2}} - \frac{6}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 8 x + 4 x^{\frac{3}{2}} - \frac{6}{5} x^{\frac{5}{2}} + C

\int (8 + x^{4})^{3} 4 x^{3} d x

\frac{d y}{d t} = 0.2 y (1000 - y)

f (x) = x 2

\frac{\partial}{\partial x} (5 x)

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