What is the integral of sqrt(1-2x)(x+2) ?

The integral of sqrt(1-2x)(x+2) is -1/5 x(-2x+1)^{3/2}-11/15 (1-2x)^{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 sqrt(1-2x)(x+2)

\int 1 - 2 x (x + 2) d x

- \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}} + C

Solution steps

\int 1 - 2 x (x + 2) d x

Expand

1 - 2 x (x + 2) : x 1 - 2 x + 2 1 - 2 x

= \int x 1 - 2 x + 2 1 - 2 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 1 - 2 x d x + \int 2 1 - 2 x d x

\int x 1 - 2 x d x = - \frac{1}{15} (3 x + 1) (- 2 x + 1)^{\frac{3}{2}}

\int 2 1 - 2 x d x = - \frac{2}{3} (1 - 2 x)^{\frac{3}{2}}

= - \frac{1}{15} (3 x + 1) (- 2 x + 1)^{\frac{3}{2}} - \frac{2}{3} (1 - 2 x)^{\frac{3}{2}}

Simplify

- \frac{1}{15} (3 x + 1) (- 2 x + 1)^{\frac{3}{2}} - \frac{2}{3} (1 - 2 x)^{\frac{3}{2}} : - \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}}

= - \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}} + C

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

\int \frac{3 x + 5}{( 2 x ^{2} + x + 1 ) ^{2}} d x

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

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

\int x ln (x) x d x

What is the integral of sqrt(1-2x)(x+2) ?
The integral of sqrt(1-2x)(x+2) is -1/5 x(-2x+1)^{3/2}-11/15 (1-2x)^{3/2}+C