What is the integral of x(x+2)^{1/2} ?

The integral of x(x+2)^{1/2} is 2(1/5 (x+2)^{5/2}-2/3 (x+2)^{3/2})+C

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integral of x(x+2)^{1/2}

\int x (x + 2)^{\frac{1}{2}} d x

2 (\frac{1}{5} (x + 2)^{\frac{5}{2}} - \frac{2}{3} (x + 2)^{\frac{3}{2}}) + C

Solution steps

\int x (x + 2)^{\frac{1}{2}} d x

Apply u-substitution

: \int 2 (u^{2} - 2) u^{2} d u

= \int 2 (u^{2} - 2) u^{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int (u^{2} - 2) u^{2} d u

Expand

(u^{2} - 2) u^{2} : u^{4} - 2 u^{2}

= 2 \cdot \int u^{4} - 2 u^{2} 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^{4} d u - \int 2 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

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

= 2 (\frac{u ^{5}}{5} - \frac{2 u ^{3}}{3})

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

= 2 (\frac{1}{5} (x + 2)^{\frac{5}{2}} - \frac{2}{3} (x + 2)^{\frac{3}{2}}) + C

\frac{d}{d x} (3 x^{2} - 7)

(2 x y + 3 y^{2}) d x = (2 x y + x^{2}) d y

x \to 2 lim (\frac{x ^{3} - 8}{2 - x})

\int \frac{u ^{2}}{u ^{2} + 1} d u

k = 1 \sum \infty \frac{k}{2 ^{k}}

What is the integral of x(x+2)^{1/2} ?
The integral of x(x+2)^{1/2} is 2(1/5 (x+2)^{5/2}-2/3 (x+2)^{3/2})+C