What is the integral of x*sqrt(2x+1) ?

The integral of x*sqrt(2x+1) is 1/15 (3x-1)(2x+1)^{3/2}+C

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integral of x*sqrt(2x+1)

\int x \cdot 2 x + 1 d x

\frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}} + C

Solution steps

\int x 2 x + 1 d x

Apply u-substitution

: \int \frac{u ( u - 1 )}{4} d u

= \int \frac{u ( u - 1 )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int u (u - 1) d u

Expand

u (u - 1) : u^{\frac{3}{2}} - u

= \frac{1}{4} \cdot \int u^{\frac{3}{2}} - u 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

= \frac{1}{4} (\int u^{\frac{3}{2}} d u - \int u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

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

= \frac{1}{4} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 2 x + 1

= \frac{1}{4} (\frac{2}{5} (2 x + 1)^{\frac{5}{2}} - \frac{2}{3} (2 x + 1)^{\frac{3}{2}})

Simplify

\frac{1}{4} (\frac{2}{5} (2 x + 1)^{\frac{5}{2}} - \frac{2}{3} (2 x + 1)^{\frac{3}{2}}) : \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}}

= \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{15} (3 x - 1) (2 x + 1)^{\frac{3}{2}} + C

x \to 0 lim (\frac{3 ( 3 - x ) - 9}{x})

f (x) = x^{\frac{9}{2}} e^{x}

(- 2, - 6), (0, 6)

\int (6 x) d x

x \to 0 lim ((1 + \frac{1}{2 x})^{x})

What is the integral of x*sqrt(2x+1) ?
The integral of x*sqrt(2x+1) is 1/15 (3x-1)(2x+1)^{3/2}+C