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

The integral of ((x^3-x^4)^2)/(sqrt(x)) is 2/13 x^{13/2}-4/15 x^{15/2}+2/17 x^{17/2}+C

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

\int \frac{( x ^{3} - x ^{4} ) ^{2}}{x} d x

\frac{2}{13} x^{\frac{13}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{17} x^{\frac{17}{2}} + C

Solution steps

\int \frac{( x ^{3} - x ^{4} ) ^{2}}{x} d x

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

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

\int \frac{( x ^{3} - x ^{4} ) ^{2} - x}{x} d x = \frac{2}{13} x^{\frac{13}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{17} x^{\frac{17}{2}} - x

\int 1 d x = x

= \frac{2}{13} x^{\frac{13}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{17} x^{\frac{17}{2}} - x + x

Simplify

= \frac{2}{13} x^{\frac{13}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{17} x^{\frac{17}{2}}

Add a constant to the solution

= \frac{2}{13} x^{\frac{13}{2}} - \frac{4}{15} x^{\frac{15}{2}} + \frac{2}{17} x^{\frac{17}{2}} + C

What is the integral of ((x^3-x^4)^2)/(sqrt(x)) ?
The integral of ((x^3-x^4)^2)/(sqrt(x)) is 2/13 x^{13/2}-4/15 x^{15/2}+2/17 x^{17/2}+C