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

The integral of x^{3/2}(x^2+3)(x+2) is 2/11 x^{11/2}+4/9 x^{9/2}+6/7 x^{7/2}+12/5 x^{5/2}+C

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integral of x^{3/2}(x^2+3)(x+2)

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

\frac{2}{11} x^{\frac{11}{2}} + \frac{4}{9} x^{\frac{9}{2}} + \frac{6}{7} x^{\frac{7}{2}} + \frac{12}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

x^{\frac{3}{2}} (x^{2} + 3) (x + 2) : x^{\frac{9}{2}} + 2 x^{\frac{7}{2}} + 3 x^{\frac{5}{2}} + 6 x^{\frac{3}{2}}

= \int x^{\frac{9}{2}} + 2 x^{\frac{7}{2}} + 3 x^{\frac{5}{2}} + 6 x^{\frac{3}{2}} 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 x^{\frac{9}{2}} d x + \int 2 x^{\frac{7}{2}} d x + \int 3 x^{\frac{5}{2}} d x + \int 6 x^{\frac{3}{2}} d x

\int x^{\frac{9}{2}} d x = \frac{2}{11} x^{\frac{11}{2}}

\int 2 x^{\frac{7}{2}} d x = \frac{4}{9} x^{\frac{9}{2}}

\int 3 x^{\frac{5}{2}} d x = \frac{6}{7} x^{\frac{7}{2}}

\int 6 x^{\frac{3}{2}} d x = \frac{12}{5} x^{\frac{5}{2}}

= \frac{2}{11} x^{\frac{11}{2}} + \frac{4}{9} x^{\frac{9}{2}} + \frac{6}{7} x^{\frac{7}{2}} + \frac{12}{5} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{11} x^{\frac{11}{2}} + \frac{4}{9} x^{\frac{9}{2}} + \frac{6}{7} x^{\frac{7}{2}} + \frac{12}{5} x^{\frac{5}{2}} + C

y^{^{'''}} + y^{^{''}} - 2 y = 0

x \to 3 lim (\frac{4 x ^{2} - 36}{2 x - 6})

x \to 1 lim (3 x^{2} - 5 x + 4)

\int_{\frac{1}{2}}^{1} x^{- 3} - 11 d x

x^{^{''}} + 4 x^{^{'}} + 4 x = t^{- 2} e^{- 2 t}

What is the integral of x^{3/2}(x^2+3)(x+2) ?
The integral of x^{3/2}(x^2+3)(x+2) is 2/11 x^{11/2}+4/9 x^{9/2}+6/7 x^{7/2}+12/5 x^{5/2}+C