What is the integral of x^2sqrt(9-x) ?

The integral of x^2sqrt(9-x) is -2/7 (9-x)^{7/2}+36/5 (9-x)^{5/2}-54(9-x)^{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^2sqrt(9-x)

\int x^{2} 9 - x d x

- \frac{2}{7} (9 - x)^{\frac{7}{2}} + \frac{36}{5} (9 - x)^{\frac{5}{2}} - 54 (9 - x)^{\frac{3}{2}} + C

Solution steps

\int x^{2} 9 - x d x

Apply u-substitution

: \int - (- u + 9)^{2} u d u

= \int - (- u + 9)^{2} u d u

Take the constant out:

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

= - \int (- u + 9)^{2} u d u

Expand

(- u + 9)^{2} u : u^{\frac{5}{2}} - 18 u^{\frac{3}{2}} + 81 u

= - \int u^{\frac{5}{2}} - 18 u^{\frac{3}{2}} + 81 u d u

Apply the Sum Rule:

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

= - (\int u^{\frac{5}{2}} d u - \int 18 u^{\frac{3}{2}} d u + \int 81 u d u)

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 18 u^{\frac{3}{2}} d u = \frac{36}{5} u^{\frac{5}{2}}

\int 81 u d u = 54 u^{\frac{3}{2}}

= - (\frac{2}{7} u^{\frac{7}{2}} - \frac{36}{5} u^{\frac{5}{2}} + 54 u^{\frac{3}{2}})

Substitute back

u = 9 - x

= - (\frac{2}{7} (9 - x)^{\frac{7}{2}} - \frac{36}{5} (9 - x)^{\frac{5}{2}} + 54 (9 - x)^{\frac{3}{2}})

Simplify

= - \frac{2}{7} (9 - x)^{\frac{7}{2}} + \frac{36}{5} (9 - x)^{\frac{5}{2}} - 54 (9 - x)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{2}{7} (9 - x)^{\frac{7}{2}} + \frac{36}{5} (9 - x)^{\frac{5}{2}} - 54 (9 - x)^{\frac{3}{2}} + C

\int 8^{3 x} d x

\int \frac{3 x ^{2}}{( 6 - 5 x ^{3} ) ^{4}} d x

\int 3 t^{3} - t d t

\int \frac{4 x ^{3}}{( x ^{4} + 7 ) ^{2}} d x

\int t^{\frac{- 1}{2}} (t^{2} - t + 2) d t

What is the integral of x^2sqrt(9-x) ?
The integral of x^2sqrt(9-x) is -2/7 (9-x)^{7/2}+36/5 (9-x)^{5/2}-54(9-x)^{3/2}+C