What is the integral of (x+7)/(sqrt(5-x)) ?

The integral of (x+7)/(sqrt(5-x)) is 2/3 (5-x)^{3/2}-24sqrt(5-x)+C

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integral of (x+7)/(sqrt(5-x))

\int \frac{x + 7}{5 - x} d x

\frac{2}{3} (5 - x)^{\frac{3}{2}} - 24 5 - x + C

Solution steps

\int \frac{x + 7}{5 - x} d x

Expand

\frac{x + 7}{5 - x} : \frac{x}{5 - x} + \frac{7}{5 - x}

= \int \frac{x}{5 - x} + \frac{7}{5 - 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}{5 - x} d x + \int \frac{7}{5 - x} d x

\int \frac{x}{5 - x} d x = - 2 (- \frac{1}{3} (5 - x)^{\frac{3}{2}} + 5 5 - x)

\int \frac{7}{5 - x} d x = - 14 5 - x

= - 2 (- \frac{1}{3} (5 - x)^{\frac{3}{2}} + 5 5 - x) - 14 5 - x

Simplify

- 2 (- \frac{1}{3} (5 - x)^{\frac{3}{2}} + 5 5 - x) - 14 5 - x : \frac{2}{3} (5 - x)^{\frac{3}{2}} - 24 5 - x

= \frac{2}{3} (5 - x)^{\frac{3}{2}} - 24 5 - x

Add a constant to the solution

= \frac{2}{3} (5 - x)^{\frac{3}{2}} - 24 5 - x + C

y^{^{'}} = x^{2} y^{3}

y = x + 4, y = \frac{( x )}{( 2 ) + 2}

f (x) = (- 16 x^{2} + 1)^{2}

\int 4 x^{2} + 9 d x

What is the integral of (x+7)/(sqrt(5-x)) ?
The integral of (x+7)/(sqrt(5-x)) is 2/3 (5-x)^{3/2}-24sqrt(5-x)+C