What is the integral of 1/(sqrt(x+1)-\sqrt{x)} ?

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

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integral of 1/(sqrt(x+1)-\sqrt{x)}

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

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

Solution steps

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

Multiply by the conjugate of

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

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

Simplify

= \int x + 1 + 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 x + 1 d x + \int x d x

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

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

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

Add a constant to the solution

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

x \to 1 lim (\frac{( x )}{x - 1})

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

\frac{d}{d x} (arccsc (- 5 x^{2}))

x \to - \infty lim (\frac{12}{x})

\int \frac{3 x - 5}{x ^{2} - 4 x + 3} d x

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