What is the integral of 5/(sqrt(x+5)+\sqrt{x)} ?

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

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integral of 5/(sqrt(x+5)+\sqrt{x)}

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{x + 5 + x} d x

Multiply by the conjugate of

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

= 5 \cdot \int \frac{1}{\frac{5}{x + 5 - x}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \frac{1}{5} \cdot \int \frac{1}{\frac{1}{x + 5 - x}} d x

Simplify

= 5 \cdot \frac{1}{5} \cdot \int x + 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

= 5 \cdot \frac{1}{5} (\int x + 5 d x - \int x d x)

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

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

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

Simplify

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

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

Add a constant to the solution

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

f (x) = x + 3, [0, 9]

x \to 2 \cdot π lim (ln (cos (x)))

\frac{s + 3}{s ^{2} + 5 s + 1}

\int (x e^{5 x}) d x

\frac{d y}{e ^{- 2 y}} = 8 x^{3} d x

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