What is the integral of (x+5)^2sqrt(3-x) ?

The integral of (x+5)^2sqrt(3-x) is -2/7 (-x+3)^{7/2}+32/5 (-x+3)^{5/2}-128/3 (-x+3)^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x+5)^2sqrt(3-x)

\int (x + 5)^{2} 3 - x d x

- \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}} + C

Solution steps

\int (x + 5)^{2} 3 - x d x

Expand

(x + 5)^{2} 3 - x : x^{2} 3 - x + 10 x 3 - x + 25 3 - x

= \int x^{2} 3 - x + 10 x 3 - x + 25 3 - x 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 x^{2} 3 - x d x + \int 10 x 3 - x d x + \int 25 3 - x d x

\int x^{2} 3 - x d x = - \frac{2}{7} (3 - x)^{\frac{7}{2}} + \frac{12}{5} (3 - x)^{\frac{5}{2}} - 6 (3 - x)^{\frac{3}{2}}

\int 10 x 3 - x d x = - 10 (- \frac{2}{5} (3 - x)^{\frac{5}{2}} + 2 (3 - x)^{\frac{3}{2}})

\int 25 3 - x d x = - \frac{50}{3} (3 - x)^{\frac{3}{2}}

= - \frac{2}{7} (3 - x)^{\frac{7}{2}} + \frac{12}{5} (3 - x)^{\frac{5}{2}} - 6 (3 - x)^{\frac{3}{2}} - 10 (- \frac{2}{5} (3 - x)^{\frac{5}{2}} + 2 (3 - x)^{\frac{3}{2}}) - \frac{50}{3} (3 - x)^{\frac{3}{2}}

Simplify

- \frac{2}{7} (3 - x)^{\frac{7}{2}} + \frac{12}{5} (3 - x)^{\frac{5}{2}} - 6 (3 - x)^{\frac{3}{2}} - 10 (- \frac{2}{5} (3 - x)^{\frac{5}{2}} + 2 (3 - x)^{\frac{3}{2}}) - \frac{50}{3} (3 - x)^{\frac{3}{2}} : - \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}}

= - \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{2}{7} (- x + 3)^{\frac{7}{2}} + \frac{32}{5} (- x + 3)^{\frac{5}{2}} - \frac{128}{3} (- x + 3)^{\frac{3}{2}} + C

What is the integral of (x+5)^2sqrt(3-x) ?
The integral of (x+5)^2sqrt(3-x) is -2/7 (-x+3)^{7/2}+32/5 (-x+3)^{5/2}-128/3 (-x+3)^{3/2}+C