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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(x)(sqrt(a)-sqrt(x))^2

\int x (a - x)^{2} d x

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

Solution steps

\int x (a - x)^{2} d x

Expand

x (a - x)^{2} : a x - 2 a x + x^{\frac{3}{2}}

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

\int a x d x = a \frac{2}{3} x^{\frac{3}{2}}

\int 2 a x d x = a x^{2}

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{2}{x ^{2} + 1})

\int_{0}^{2} 2 π (5 - x) (8 - x^{3}) d x

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

\frac{x d y}{d x} + 5 y = 7 x^{2}, f (2) = 5

\frac{d}{d x} (C x e^{- x})

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