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

The integral of x(2x+1)sqrt(2x+1) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(2x+1)sqrt(2x+1)

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

- 2 x^{3} 2 x + 1 + \frac{9}{7} x^{2} (2 x + 1)^{\frac{3}{2}} + x^{2} 2 x + 1 - \frac{1}{3} x 2 x + 1 - \frac{18}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{1}{5} (2 x + 1)^{\frac{5}{2}} - \frac{2}{3} 2 x + 1 + \frac{88}{105} (2 x + 1)^{\frac{3}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{( 3 x + 1 ) ( x ^{2} + x )}{2 x + 1} d x = 3 x^{3} 2 x + 1 - \frac{9}{7} x^{2} (2 x + 1)^{\frac{3}{2}} + \frac{1}{3} x 2 x + 1 + \frac{18}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{1}{5} (2 x + 1)^{\frac{5}{2}} - \frac{88}{105} (2 x + 1)^{\frac{3}{2}} + \frac{2}{3} 2 x + 1

= x (x^{2} + x) 2 x + 1 - (3 x^{3} 2 x + 1 - \frac{9}{7} x^{2} (2 x + 1)^{\frac{3}{2}} + \frac{1}{3} x 2 x + 1 + \frac{18}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{1}{5} (2 x + 1)^{\frac{5}{2}} - \frac{88}{105} (2 x + 1)^{\frac{3}{2}} + \frac{2}{3} 2 x + 1)

Simplify

x (x^{2} + x) 2 x + 1 - (3 x^{3} 2 x + 1 - \frac{9}{7} x^{2} (2 x + 1)^{\frac{3}{2}} + \frac{1}{3} x 2 x + 1 + \frac{18}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{1}{5} (2 x + 1)^{\frac{5}{2}} - \frac{88}{105} (2 x + 1)^{\frac{3}{2}} + \frac{2}{3} 2 x + 1) : - 2 x^{3} 2 x + 1 + \frac{9}{7} x^{2} (2 x + 1)^{\frac{3}{2}} + x^{2} 2 x + 1 - \frac{1}{3} x 2 x + 1 - \frac{18}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{1}{5} (2 x + 1)^{\frac{5}{2}} - \frac{2}{3} 2 x + 1 + \frac{88}{105} (2 x + 1)^{\frac{3}{2}}

= - 2 x^{3} 2 x + 1 + \frac{9}{7} x^{2} (2 x + 1)^{\frac{3}{2}} + x^{2} 2 x + 1 - \frac{1}{3} x 2 x + 1 - \frac{18}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{1}{5} (2 x + 1)^{\frac{5}{2}} - \frac{2}{3} 2 x + 1 + \frac{88}{105} (2 x + 1)^{\frac{3}{2}}

Add a constant to the solution

= - 2 x^{3} 2 x + 1 + \frac{9}{7} x^{2} (2 x + 1)^{\frac{3}{2}} + x^{2} 2 x + 1 - \frac{1}{3} x 2 x + 1 - \frac{18}{35} x (2 x + 1)^{\frac{3}{2}} - \frac{1}{5} (2 x + 1)^{\frac{5}{2}} - \frac{2}{3} 2 x + 1 + \frac{88}{105} (2 x + 1)^{\frac{3}{2}} + C

\int \frac{x ^{2} + 3 x - 12}{x ^{3} + 3 x ^{2} - 28 x} d x

\int \frac{6 x ^{2} + 3 x + 6}{x ^{3} - 1} d x

\int (sec (θ) - tan (θ))^{2} d θ

\int 2 u e^{u} d u

\int \frac{x ^{8} + 24}{x ^{7}} d x

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