What is the integral of ((1-2x)^2)/(sqrt(2x)) ?

The integral of ((1-2x)^2)/(sqrt(2x)) is sqrt(2)sqrt(x)-(4sqrt(2))/3 x^{3/2}+(4sqrt(2))/5 x^{5/2}+C

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integral of ((1-2x)^2)/(sqrt(2x))

\int \frac{( 1 - 2 x ) ^{2}}{2 x} d x

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

Solution steps

\int \frac{( 1 - 2 x ) ^{2}}{2 x} d x

Expand

\frac{( 1 - 2 x ) ^{2}}{2 x} : \frac{1}{2 x} - 22 x + 22 x^{\frac{3}{2}}

= \int \frac{1}{2 x} - 22 x + 22 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{1}{2 x} d x - \int 22 x d x + \int 22 x^{\frac{3}{2}} d x

\int \frac{1}{2 x} d x = 2 x

\int 22 x d x = \frac{4 2}{3} x^{\frac{3}{2}}

\int 22 x^{\frac{3}{2}} d x = \frac{4 2}{5} x^{\frac{5}{2}}

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

Add a constant to the solution

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

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What is the integral of ((1-2x)^2)/(sqrt(2x)) ?
The integral of ((1-2x)^2)/(sqrt(2x)) is sqrt(2)sqrt(x)-(4sqrt(2))/3 x^{3/2}+(4sqrt(2))/5 x^{5/2}+C