What is the integral of x^5sqrt(1-x^3) ?

The integral of x^5sqrt(1-x^3) is -1/45 (-6(-x^3+1)^{5/2}+10(-x^3+1)^{3/2})+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^5sqrt(1-x^3)

\int x^{5} 1 - x^{3} d x

- \frac{1}{45} (- 6 (- x^{3} + 1)^{\frac{5}{2}} + 10 (- x^{3} + 1)^{\frac{3}{2}}) + C

Solution steps

\int x^{5} 1 - x^{3} d x

Apply u-substitution

: \int \frac{u ^{\frac{1}{5}} - u ^{\frac{3}{5}} + 1}{5} d u

= \int \frac{u ^{\frac{1}{5}} - u ^{\frac{3}{5}} + 1}{5} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int u^{\frac{1}{5}} - u^{\frac{3}{5}} + 1 d u

Apply u-substitution

: \int - \frac{5 ( - v + 1 ) v}{3} d v

= \frac{1}{5} \cdot \int - \frac{5 ( - v + 1 ) v}{3} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} (- \frac{5}{3} \cdot \int (- v + 1) v d v)

Expand

(- v + 1) v : - v^{\frac{3}{2}} + v

= \frac{1}{5} (- \frac{5}{3} \cdot \int - v^{\frac{3}{2}} + v d v)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{5} (- \frac{5}{3} (- \int v^{\frac{3}{2}} d v + \int v d v))

\int v^{\frac{3}{2}} d v = \frac{2}{5} v^{\frac{5}{2}}

\int v d v = \frac{2}{3} v^{\frac{3}{2}}

= \frac{1}{5} (- \frac{5}{3} (- \frac{2}{5} v^{\frac{5}{2}} + \frac{2}{3} v^{\frac{3}{2}}))

Substitute back

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

Simplify

\frac{1}{5} (- \frac{5}{3} (- \frac{2}{5} (- (x^{5})^{\frac{3}{5}} + 1)^{\frac{5}{2}} + \frac{2}{3} (- (x^{5})^{\frac{3}{5}} + 1)^{\frac{3}{2}})) : - \frac{1}{45} (- 6 (- x^{3} + 1)^{\frac{5}{2}} + 10 (- x^{3} + 1)^{\frac{3}{2}})

= - \frac{1}{45} (- 6 (- x^{3} + 1)^{\frac{5}{2}} + 10 (- x^{3} + 1)^{\frac{3}{2}})

Add a constant to the solution

= - \frac{1}{45} (- 6 (- x^{3} + 1)^{\frac{5}{2}} + 10 (- x^{3} + 1)^{\frac{3}{2}}) + C

\int \frac{y ^{3} - 4 y + 3}{y + 5} d y

\int \frac{1}{x + 2 x ^{3}} d x

\int \frac{1}{2 x ^{2} x ^{2} - 16} d x

\int \frac{7}{w ^{2} + 3 w + 3} d w

\int 4 cot^{3} (x) csc (x) d x

What is the integral of x^5sqrt(1-x^3) ?
The integral of x^5sqrt(1-x^3) is -1/45 (-6(-x^3+1)^{5/2}+10(-x^3+1)^{3/2})+C