What is the integral of xsqrt(2x+3) ?

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

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integral of xsqrt(2x+3)

\int x 2 x + 3 d x

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

Solution steps

\int x 2 x + 3 d x

Apply u-substitution

: \int \frac{u ( u - 3 )}{4} d u

= \int \frac{u ( u - 3 )}{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 - 3) d u

Expand

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

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

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

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

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

Substitute back

u = 2 x + 3

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (cos^{6} (x^{9} y^{5}))

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

y^{^{''}} + 2 y^{^{'}} + 2 y = 8 e^{2 t}, y (0) = 0, y^{^{'}} (0) = 1

\frac{d y}{d t} = 0.5 y, y (2) = 100

\int \frac{x ^{2}}{x ^{3} - 5 x} d x

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