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

The integral of (sqrt(x))/(sqrt(x)-5) is x+10sqrt(x)+50ln|sqrt(x)-5|-75+C

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

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

x + 10 x + 50 ln x - 5 - 75 + C

Solution steps

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

Apply u-substitution

: \int \frac{2 ( u ^{2} + 10 u + 25 )}{u} d u

= \int \frac{2 ( u ^{2} + 10 u + 25 )}{u} 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} + 10 u + 25}{u} d u

Expand

\frac{u ^{2} + 10 u + 25}{u} : u + 10 + \frac{25}{u}

= 2 \cdot \int u + 10 + \frac{25}{u} d u

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 u d u + \int 10 d u + \int \frac{25}{u} d u)

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

\int 10 d u = 10 u

\int \frac{25}{u} d u = 25 ln ∣ u ∣

= 2 (\frac{u ^{2}}{2} + 10 u + 25 ln ∣ u ∣)

Substitute back

u = x - 5

= 2 (\frac{( x - 5 ) ^{2}}{2} + 10 (x - 5) + 25 ln x - 5)

Simplify

2 (\frac{( x - 5 ) ^{2}}{2} + 10 (x - 5) + 25 ln x - 5) : x + 10 x + 50 ln x - 5 - 75

= x + 10 x + 50 ln x - 5 - 75

Add a constant to the solution

= x + 10 x + 50 ln x - 5 - 75 + C

\int arctan (y) d y

f (x) = \frac{1}{2 x}

x \to 2 lim ((x + 1) (3 x^{2} - 9))

2 x^{2} + 5, [- 2, 3]

x \to 1 lim (\frac{∣ 1 - x ∣}{1 - x})

What is the integral of (sqrt(x))/(sqrt(x)-5) ?
The integral of (sqrt(x))/(sqrt(x)-5) is x+10sqrt(x)+50ln|sqrt(x)-5|-75+C