What is the integral of (11-x)/(sqrt(14-x)) ?

The integral of (11-x)/(sqrt(14-x)) is -2/3 (-x+14)^{3/2}+6sqrt(-x+14)+C

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integral of (11-x)/(sqrt(14-x))

\int \frac{11 - x}{14 - x} d x

- \frac{2}{3} (- x + 14)^{\frac{3}{2}} + 6 - x + 14 + C

Solution steps

\int \frac{11 - x}{14 - x} d x

Expand

\frac{11 - x}{14 - x} : \frac{11}{14 - x} - \frac{x}{14 - x}

= \int \frac{11}{14 - x} - \frac{x}{14 - 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{11}{14 - x} d x - \int \frac{x}{14 - x} d x

\int \frac{11}{14 - x} d x = - 22 14 - x

\int \frac{x}{14 - x} d x = - 2 (- \frac{1}{3} (14 - x)^{\frac{3}{2}} + 14 14 - x)

= - 22 14 - x - (- 2 (- \frac{1}{3} (14 - x)^{\frac{3}{2}} + 14 14 - x))

Simplify

- 22 14 - x - (- 2 (- \frac{1}{3} (14 - x)^{\frac{3}{2}} + 14 14 - x)) : - \frac{2}{3} (- x + 14)^{\frac{3}{2}} + 6 - x + 14

= - \frac{2}{3} (- x + 14)^{\frac{3}{2}} + 6 - x + 14

Add a constant to the solution

= - \frac{2}{3} (- x + 14)^{\frac{3}{2}} + 6 - x + 14 + C

What is the integral of (11-x)/(sqrt(14-x)) ?
The integral of (11-x)/(sqrt(14-x)) is -2/3 (-x+14)^{3/2}+6sqrt(-x+14)+C