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

The integral of x/(3+sqrt(x)) is 2/3 x^{3/2}-3x+18sqrt(x)-54ln|sqrt(x)+3|+C

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

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

\frac{2}{3} x^{\frac{3}{2}} - 3 x + 18 x - 54 ln x + 3 + C

Solution steps

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

Apply u-substitution

: \int \frac{2 u ^{3}}{3 + u} d u

= \int \frac{2 u ^{3}}{3 + 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 ^{3}}{3 + u} d u

Long division

\frac{u ^{3}}{3 + u} : u^{2} - 3 u + 9 - \frac{27}{u + 3}

= 2 \cdot \int u^{2} - 3 u + 9 - \frac{27}{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

= 2 (\int u^{2} d u - \int 3 u d u + \int 9 d u - \int \frac{27}{u + 3} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 3 u d u = \frac{3 u ^{2}}{2}

\int 9 d u = 9 u

\int \frac{27}{u + 3} d u = 27 ln ∣ u + 3 ∣

= 2 (\frac{u ^{3}}{3} - \frac{3 u ^{2}}{2} + 9 u - 27 ln ∣ u + 3 ∣)

Substitute back

u = x

= 2 (\frac{( x ) ^{3}}{3} - \frac{3 ( x ) ^{2}}{2} + 9 x - 27 ln x + 3)

Simplify

2 (\frac{( x ) ^{3}}{3} - \frac{3 ( x ) ^{2}}{2} + 9 x - 27 ln x + 3) : \frac{2}{3} x^{\frac{3}{2}} - 3 x + 18 x - 54 ln x + 3

= \frac{2}{3} x^{\frac{3}{2}} - 3 x + 18 x - 54 ln x + 3

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} - 3 x + 18 x - 54 ln x + 3 + C

\frac{e ^{2}}{x ^{2} - 2 x}

\frac{( 2 )}{sin ( x ) + cos ( x )}

\frac{d}{d x} (\frac{2 x ^{2}}{3 y ^{2}})

\frac{d}{d x} (1 - 3 x^{2})

x \to 0 + lim (tan (x) \cdot ln (x))

What is the integral of x/(3+sqrt(x)) ?
The integral of x/(3+sqrt(x)) is 2/3 x^{3/2}-3x+18sqrt(x)-54ln|sqrt(x)+3|+C