What is the integral of (s(s+1)^2)/(sqrt(s)) ?

The integral of (s(s+1)^2)/(sqrt(s)) is 2/7 s^{7/2}+4/5 s^{5/2}+2/3 s^{3/2}+C

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integral of (s(s+1)^2)/(sqrt(s))

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

\frac{2}{7} s^{\frac{7}{2}} + \frac{4}{5} s^{\frac{5}{2}} + \frac{2}{3} s^{\frac{3}{2}} + C

Solution steps

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

Expand

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

= \int s^{\frac{5}{2}} + 2 s^{\frac{3}{2}} + s d s

Apply the Sum Rule:

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

= \int s^{\frac{5}{2}} d s + \int 2 s^{\frac{3}{2}} d s + \int s d s

\int s^{\frac{5}{2}} d s = \frac{2}{7} s^{\frac{7}{2}}

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

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

= \frac{2}{7} s^{\frac{7}{2}} + \frac{4}{5} s^{\frac{5}{2}} + \frac{2}{3} s^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} s^{\frac{7}{2}} + \frac{4}{5} s^{\frac{5}{2}} + \frac{2}{3} s^{\frac{3}{2}} + C

(sin (x) cos (x))^{^{'}}

\int_{1}^{2} 5 x d x

f (x) = 2^{x} 1

\frac{d y}{d t} - 2 t y = 3 t^{2} e^{t^{2}}

What is the integral of (s(s+1)^2)/(sqrt(s)) ?
The integral of (s(s+1)^2)/(sqrt(s)) is 2/7 s^{7/2}+4/5 s^{5/2}+2/3 s^{3/2}+C