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

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

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

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

Solution steps

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

Expand

(x + 1) (x^{3} - 1) : x x^{3} - x + x^{3} - 1

= \int x x^{3} - x + x^{3} - 1 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 x^{3} d x - \int x d x + \int x^{3} d x - \int 1 d x

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

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

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

\int 1 d x = x

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

Add a constant to the solution

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

\int \frac{6}{4 - 3 x} d x

\int 5 t cos (π t) d t

\int sin^{4} (6 x) cos (6 x) d x

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

\int x (7 - x)^{5} d x

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