What is the integral of (sqrt(x))/(sqrt(x)-7) ?

The integral of (sqrt(x))/(sqrt(x)-7) is x+14sqrt(x)+98ln|sqrt(x)-7|-147+C

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integral of (sqrt(x))/(sqrt(x)-7)

\int \frac{x}{x - 7} d x

x + 14 x + 98 ln x - 7 - 147 + C

Solution steps

\int \frac{x}{x - 7} d x

Apply u-substitution

: \int \frac{2 ( u ^{2} + 14 u + 49 )}{u} d u

= \int \frac{2 ( u ^{2} + 14 u + 49 )}{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} + 14 u + 49}{u} d u

Expand

\frac{u ^{2} + 14 u + 49}{u} : u + 14 + \frac{49}{u}

= 2 \cdot \int u + 14 + \frac{49}{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 14 d u + \int \frac{49}{u} d u)

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

\int 14 d u = 14 u

\int \frac{49}{u} d u = 49 ln ∣ u ∣

= 2 (\frac{u ^{2}}{2} + 14 u + 49 ln ∣ u ∣)

Substitute back

u = x - 7

= 2 (\frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln x - 7)

Simplify

2 (\frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln x - 7) : x + 14 x + 98 ln x - 7 - 147

= x + 14 x + 98 ln x - 7 - 147

Add a constant to the solution

= x + 14 x + 98 ln x - 7 - 147 + C

x^{3} - 3 x + 1

\int \frac{3 x ^{2}}{x ^{3} + 7} d x

\frac{\partial}{\partial x} (ln (cos (x)))

\frac{( x ^{2} )}{x ^{2} + 27}

\frac{d}{d x} (\frac{cos ( x )}{lo g _{3} ( x )})

What is the integral of (sqrt(x))/(sqrt(x)-7) ?
The integral of (sqrt(x))/(sqrt(x)-7) is x+14sqrt(x)+98ln|sqrt(x)-7|-147+C