What is the integral of ((1-x)^3)/(x^3sqrt(x)) ?

The integral of ((1-x)^3)/(x^3sqrt(x)) is -2/(5x^{5/2)}+2/(x^{3/2)}-6/(sqrt(x))-2sqrt(x)+C

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integral of ((1-x)^3)/(x^3sqrt(x))

\int \frac{( 1 - x ) ^{3}}{x ^{3} x} d x

- \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x + C

Solution steps

\int \frac{( 1 - x ) ^{3}}{x ^{3} x} d x

Expand

\frac{( 1 - x ) ^{3}}{x ^{3} x} : \frac{1}{x ^{\frac{7}{2}}} - \frac{3}{x ^{\frac{5}{2}}} + \frac{3}{x ^{\frac{3}{2}}} - \frac{1}{x ^{\frac{1}{2}}}

= \int \frac{1}{x ^{\frac{7}{2}}} - \frac{3}{x ^{\frac{5}{2}}} + \frac{3}{x ^{\frac{3}{2}}} - \frac{1}{x ^{\frac{1}{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{1}{x ^{\frac{7}{2}}} d x - \int \frac{3}{x ^{\frac{5}{2}}} d x + \int \frac{3}{x ^{\frac{3}{2}}} d x - \int \frac{1}{x ^{\frac{1}{2}}} d x

\int \frac{1}{x ^{\frac{7}{2}}} d x = - \frac{2}{5 x ^{\frac{5}{2}}}

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

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

\int \frac{1}{x ^{\frac{1}{2}}} d x = 2 x^{\frac{1}{2}}

= - \frac{2}{5 x ^{\frac{5}{2}}} - (- \frac{2}{x ^{\frac{3}{2}}}) - \frac{6}{x} - 2 x^{\frac{1}{2}}

Simplify

- \frac{2}{5 x ^{\frac{5}{2}}} - (- \frac{2}{x ^{\frac{3}{2}}}) - \frac{6}{x} - 2 x^{\frac{1}{2}} : - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x

= - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x

Add a constant to the solution

= - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x + C

\frac{d}{d t} (e^{3 t} cos (2 t))

f (x) = ln (4 - x^{8})

\frac{d ^{2} y}{d x ^{2}} - 49 \frac{d y}{d x} = e^{7 x}

\int_{0}^{1} x cos (2 π x) d x

- 2 x + 5

What is the integral of ((1-x)^3)/(x^3sqrt(x)) ?
The integral of ((1-x)^3)/(x^3sqrt(x)) is -2/(5x^{5/2)}+2/(x^{3/2)}-6/(sqrt(x))-2sqrt(x)+C