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

The integral of x/(x-sqrt(x)) is x+2sqrt(x)+2ln|sqrt(x)-1|-3+C

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

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

x + 2 x + 2 ln x - 1 - 3 + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \int \frac{( v + 1 ) ^{2}}{v} d v

Expand

\frac{( v + 1 ) ^{2}}{v} : v + 2 + \frac{1}{v}

= 2 \cdot \int v + 2 + \frac{1}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int v d v + \int 2 d v + \int \frac{1}{v} d v)

\int v d v = \frac{v ^{2}}{2}

\int 2 d v = 2 v

\int \frac{1}{v} d v = ln ∣ v ∣

= 2 (\frac{v ^{2}}{2} + 2 v + ln ∣ v ∣)

Substitute back

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

Simplify

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

= x + 2 x + 2 ln x - 1 - 3

Add a constant to the solution

= x + 2 x + 2 ln x - 1 - 3 + C

\frac{d}{d x} (e^{y^{2} - x^{2}})

y^{^{''}} + y^{^{'}} + 4 y = 0, y (0) = 0, y^{^{'}} (0) = 0

\int sec^{3} (5 x) tan^{5} (5 x) d x

f (x) = 16 x, at x = 4

\int_{3}^{2} (5 x^{2} + 4 x + 9) d x

What is the integral of x/(x-sqrt(x)) ?
The integral of x/(x-sqrt(x)) is x+2sqrt(x)+2ln|sqrt(x)-1|-3+C