What is the integral of x/((x+5)sqrt(x+2)) ?

The integral of x/((x+5)sqrt(x+2)) is 2sqrt(x+2)-(10)/(sqrt(3))arctan(sqrt(1/3 (x+2)))+C

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integral of x/((x+5)sqrt(x+2))

\int \frac{x}{( x + 5 ) x + 2} d x

2 x + 2 - \frac{10}{3} arctan (\frac{1}{3} (x + 2)) + C

Solution steps

\int \frac{x}{( x + 5 ) x + 2} d x

Apply u-substitution

= \int \frac{u - 5}{u u - 3} d u

Expand

\frac{u - 5}{u u - 3} : \frac{1}{u - 3} - \frac{5}{u u - 3}

= \int \frac{1}{u - 3} - \frac{5}{u u - 3} 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

= \int \frac{1}{u - 3} d u - \int \frac{5}{u u - 3} d u

\int \frac{1}{u - 3} d u = 2 u - 3

\int \frac{5}{u u - 3} d u = \frac{10}{3} arctan (\frac{1}{3} (u - 3))

= 2 u - 3 - \frac{10}{3} arctan (\frac{1}{3} (u - 3))

Substitute back

u = x + 5

= 2 x + 5 - 3 - \frac{10}{3} arctan (\frac{1}{3} (x + 5 - 3))

Simplify

2 x + 5 - 3 - \frac{10}{3} arctan (\frac{1}{3} (x + 5 - 3)) : 2 x + 2 - \frac{10}{3} arctan (\frac{1}{3} (x + 2))

= 2 x + 2 - \frac{10}{3} arctan (\frac{1}{3} (x + 2))

Add a constant to the solution

= 2 x + 2 - \frac{10}{3} arctan (\frac{1}{3} (x + 2)) + C

What is the integral of x/((x+5)sqrt(x+2)) ?
The integral of x/((x+5)sqrt(x+2)) is 2sqrt(x+2)-(10)/(sqrt(3))arctan(sqrt(1/3 (x+2)))+C