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

The integral of (sqrt(x))/(sqrt(x-1)) is sqrt(x/(x-1))(x-1)+ln|sqrt(x)+sqrt(x-1)|+C

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

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

\frac{x}{x - 1} (x - 1) + ln x + x - 1 + C

Solution steps

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

Simplify

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

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

Apply u-substitution

: \int \frac{u + 1}{u} d u

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

Apply Integration By Parts

: \frac{u + 1}{u} u - \int - \frac{1}{2 u ^{\frac{1}{2}} u + 1} d u

= \frac{u + 1}{u} u - \int - \frac{1}{2 u ^{\frac{1}{2}} u + 1} d u

\int - \frac{1}{2 u ^{\frac{1}{2}} u + 1} d u = - ln u + 1 + u

= \frac{u + 1}{u} u - (- ln u + 1 + u)

Substitute back

u = x - 1

= \frac{x - 1 + 1}{x - 1} (x - 1) - (- ln x - 1 + 1 + x - 1)

Simplify

= \frac{x}{x - 1} (x - 1) + ln x + x - 1

Add a constant to the solution

= \frac{x}{x - 1} (x - 1) + ln x + x - 1 + C

(\frac{( 3 x ^{2} )}{2 x + 4})^{3}

x \to 0 lim (x \cdot ln^{2} (x))

x \to 0 lim (\frac{3 e ^{x} - 3}{x})

x \to \infty lim (- 5 x^{6} - 6 x^{4} - x^{3} + 7 x)

\int \frac{( ln ( x ^{9} ) )}{x} d x

What is the integral of (sqrt(x))/(sqrt(x-1)) ?
The integral of (sqrt(x))/(sqrt(x-1)) is sqrt(x/(x-1))(x-1)+ln|sqrt(x)+sqrt(x-1)|+C