What is the integral of (2+3x^2+4x^3)/(sqrt(x)) ?

The integral of (2+3x^2+4x^3)/(sqrt(x)) is 4sqrt(x)+6/5 x^{5/2}+8/7 x^{7/2}+C

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integral of (2+3x^2+4x^3)/(sqrt(x))

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

4 x + \frac{6}{5} x^{\frac{5}{2}} + \frac{8}{7} x^{\frac{7}{2}} + C

Solution steps

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

Expand

\frac{2 + 3 x ^{2} + 4 x ^{3}}{x} : \frac{2}{x} + 3 x^{\frac{3}{2}} + 4 x^{\frac{5}{2}}

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

\int \frac{2}{x} d x = 4 x

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

\int 4 x^{\frac{5}{2}} d x = \frac{8}{7} x^{\frac{7}{2}}

= 4 x + \frac{6}{5} x^{\frac{5}{2}} + \frac{8}{7} x^{\frac{7}{2}}

Add a constant to the solution

= 4 x + \frac{6}{5} x^{\frac{5}{2}} + \frac{8}{7} x^{\frac{7}{2}} + C

What is the integral of (2+3x^2+4x^3)/(sqrt(x)) ?
The integral of (2+3x^2+4x^3)/(sqrt(x)) is 4sqrt(x)+6/5 x^{5/2}+8/7 x^{7/2}+C