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

The integral of ((2x+3)^2)/(sqrt(x)) is 8/5 x^{5/2}+8x^{3/2}+18sqrt(x)+C

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

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

\frac{8}{5} x^{\frac{5}{2}} + 8 x^{\frac{3}{2}} + 18 x + C

Solution steps

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

Expand

\frac{( 2 x + 3 ) ^{2}}{x} : 4 x^{\frac{3}{2}} + 12 x + \frac{9}{x}

= \int 4 x^{\frac{3}{2}} + 12 x + \frac{9}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 4 x^{\frac{3}{2}} d x + \int 12 x d x + \int \frac{9}{x} d x

\int 4 x^{\frac{3}{2}} d x = \frac{8}{5} x^{\frac{5}{2}}

\int 12 x d x = 8 x^{\frac{3}{2}}

\int \frac{9}{x} d x = 18 x

= \frac{8}{5} x^{\frac{5}{2}} + 8 x^{\frac{3}{2}} + 18 x

Add a constant to the solution

= \frac{8}{5} x^{\frac{5}{2}} + 8 x^{\frac{3}{2}} + 18 x + C

\frac{d}{d x} (2 sin (x))

\frac{2}{1 - t}, at t = - 2

\frac{d}{d x} (2 x ln (2 x) + x)

\frac{d y}{d x} - \frac{2 x}{2 - x ^{2}} \cdot y = e^{x}, y (2) = 4

f (x) = 3 0^{3} - 3 x^{2} + 12 x - 74

What is the integral of ((2x+3)^2)/(sqrt(x)) ?
The integral of ((2x+3)^2)/(sqrt(x)) is 8/5 x^{5/2}+8x^{3/2}+18sqrt(x)+C