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

The integral of (5+sqrt(x))/(5-sqrt(x)) is -x-20sqrt(x)-100ln|-sqrt(x)+5|+C

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

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

- x - 20 x - 100 ln - x + 5 + C

Solution steps

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

Apply u-substitution

= \int \frac{2 u ( 5 + u )}{5 - 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 ( 5 + u )}{5 - u} d u

Long division

\frac{u ( 5 + u )}{5 - u} : - u - 10 + \frac{50}{- u + 5}

= 2 \cdot \int - u - 10 + \frac{50}{- u + 5} 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{50}{- u + 5} d u)

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

\int 10 d u = 10 u

\int \frac{50}{- u + 5} d u = - 50 ln ∣ - u + 5 ∣

= 2 (- \frac{u ^{2}}{2} - 10 u - 50 ln ∣ - u + 5 ∣)

Substitute back

u = x

= 2 (- \frac{( x ) ^{2}}{2} - 10 x - 50 ln - x + 5)

Simplify

2 (- \frac{( x ) ^{2}}{2} - 10 x - 50 ln - x + 5) : - x - 20 x - 100 ln - x + 5

= - x - 20 x - 100 ln - x + 5

Add a constant to the solution

= - x - 20 x - 100 ln - x + 5 + C

\int \frac{e ^{x} + 1}{x} d x

t an g e n t f (x) = \frac{4}{x + 1}, at x = 1

x \to 25 lim (x - 25)

(x + 1) d y = (x + 6) d x

\frac{\partial}{\partial x} (x^{2} + x)

What is the integral of (5+sqrt(x))/(5-sqrt(x)) ?
The integral of (5+sqrt(x))/(5-sqrt(x)) is -x-20sqrt(x)-100ln|-sqrt(x)+5|+C