What is the integral of (sqrt(x)-5)/(sqrt(x)+3) ?

The integral of (sqrt(x)-5)/(sqrt(x)+3) is x-16sqrt(x)+48ln|sqrt(x)+3|-57+C

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integral of (sqrt(x)-5)/(sqrt(x)+3)

\int \frac{x - 5}{x + 3} d x

x - 16 x + 48 ln x + 3 - 57 + C

Solution steps

\int \frac{x - 5}{x + 3} d x

Expand

\frac{x - 5}{x + 3} : \frac{x}{x + 3} - \frac{5}{x + 3}

Apply the Sum Rule:

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

= \int \frac{x}{x + 3} d x - \int \frac{5}{x + 3} d x

\int \frac{x}{x + 3} d x = x - 6 x + 18 ln x + 3 - 27

\int \frac{5}{x + 3} d x = 10 (x + 3 - 3 ln x + 3)

= x - 6 x + 18 ln x + 3 - 27 - 10 (x + 3 - 3 ln x + 3)

Simplify

x - 6 x + 18 ln x + 3 - 27 - 10 (x + 3 - 3 ln x + 3) : x - 16 x + 48 ln x + 3 - 57

= x - 16 x + 48 ln x + 3 - 57

Add a constant to the solution

= x - 16 x + 48 ln x + 3 - 57 + C

y = (x - 2)^{2}, y = x

f (x) = arcsin (3 x)

f (x) = 2 x (x^{2} + 1)

\int 7 x (x^{2} + 2)^{2} d x

\int cot (x) cos (x) d x

What is the integral of (sqrt(x)-5)/(sqrt(x)+3) ?
The integral of (sqrt(x)-5)/(sqrt(x)+3) is x-16sqrt(x)+48ln|sqrt(x)+3|-57+C