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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

3 - 2 t + 4 cos (2 t)

\frac{d}{d x} (x cos^{2} (x))

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

\int \frac{1}{1 + 2 x} d x

x \to 3 lim (4 x)

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