What is the integral of (x+1)/(sqrt(3-x)) ?

The integral of (x+1)/(sqrt(3-x)) is 2/3 (3-x)^{3/2}-8sqrt(3-x)+C

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integral of (x+1)/(sqrt(3-x))

\int \frac{x + 1}{3 - x} d x

\frac{2}{3} (3 - x)^{\frac{3}{2}} - 8 3 - x + C

Solution steps

\int \frac{x + 1}{3 - x} d x

Expand

\frac{x + 1}{3 - x} : \frac{x}{3 - x} + \frac{1}{3 - x}

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

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

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

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

Simplify

- 2 (- \frac{1}{3} (3 - x)^{\frac{3}{2}} + 3 3 - x) - 2 3 - x : \frac{2}{3} (3 - x)^{\frac{3}{2}} - 8 3 - x

= \frac{2}{3} (3 - x)^{\frac{3}{2}} - 8 3 - x

Add a constant to the solution

= \frac{2}{3} (3 - x)^{\frac{3}{2}} - 8 3 - x + C

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What is the integral of (x+1)/(sqrt(3-x)) ?
The integral of (x+1)/(sqrt(3-x)) is 2/3 (3-x)^{3/2}-8sqrt(3-x)+C