What is the integral of ((2x+3))/(sqrt(x^2+3x)) ?

The integral of ((2x+3))/(sqrt(x^2+3x)) is 2sqrt(x^2+3x)+C

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integral of ((2x+3))/(sqrt(x^2+3x))

\int \frac{( 2 x + 3 )}{x ^{2} + 3 x} d x

2 x^{2} + 3 x + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 2 u^{\frac{1}{2}}

Substitute back

u = x^{2} + 3 x

= 2 (x^{2} + 3 x)^{\frac{1}{2}}

Apply radical rule:

= 2 x^{2} + 3 x

Add a constant to the solution

= 2 x^{2} + 3 x + C

ln (sec (3 x) + tan (3 x))

\int 2 e^{- 2 x} sin (x) d x

\frac{\partial}{\partial y} (e^{x} ln (y))

\frac{d}{d y} (y sin (x y))

x^{2} + 4, - x^{2} - 1, [0, 1]

What is the integral of ((2x+3))/(sqrt(x^2+3x)) ?
The integral of ((2x+3))/(sqrt(x^2+3x)) is 2sqrt(x^2+3x)+C