What is the integral of (12-x)/(sqrt(x)) ?

The integral of (12-x)/(sqrt(x)) is 24sqrt(x)-2/3 x^{3/2}+C

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integral of (12-x)/(sqrt(x))

\int \frac{12 - x}{x} d x

24 x - \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

\int \frac{12 - x}{x} d x

Expand

\frac{12 - x}{x} : \frac{12}{x} - x

= \int \frac{12}{x} - 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 \frac{12}{x} d x - \int x d x

\int \frac{12}{x} d x = 24 x

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

= 24 x - \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= 24 x - \frac{2}{3} x^{\frac{3}{2}} + C

What is the integral of (12-x)/(sqrt(x)) ?
The integral of (12-x)/(sqrt(x)) is 24sqrt(x)-2/3 x^{3/2}+C