What is the integral of (3x^2+4)sqrt(x-1) ?

The integral of (3x^2+4)sqrt(x-1) is 6/7 x^2(x-1)^{3/2}+24/35 x(x-1)^{3/2}+328/105 (x-1)^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (3x^2+4)sqrt(x-1)

\int (3 x^{2} + 4) x - 1 d x

\frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}} + C

Solution steps

\int (3 x^{2} + 4) x - 1 d x

Apply Integration By Parts

= 2 x^{2} (x - 1)^{\frac{3}{2}} + \frac{8}{3} (x - 1)^{\frac{3}{2}} - \int 4 x (x - 1)^{\frac{3}{2}} d x

\int 4 x (x - 1)^{\frac{3}{2}} d x = 4 (\frac{2}{7} x^{2} (x - 1)^{\frac{3}{2}} - \frac{6}{35} x (x - 1)^{\frac{3}{2}} - \frac{4}{35} (x - 1)^{\frac{3}{2}})

= 2 x^{2} (x - 1)^{\frac{3}{2}} + \frac{8}{3} (x - 1)^{\frac{3}{2}} - 4 (\frac{2}{7} x^{2} (x - 1)^{\frac{3}{2}} - \frac{6}{35} x (x - 1)^{\frac{3}{2}} - \frac{4}{35} (x - 1)^{\frac{3}{2}})

Simplify

2 x^{2} (x - 1)^{\frac{3}{2}} + \frac{8}{3} (x - 1)^{\frac{3}{2}} - 4 (\frac{2}{7} x^{2} (x - 1)^{\frac{3}{2}} - \frac{6}{35} x (x - 1)^{\frac{3}{2}} - \frac{4}{35} (x - 1)^{\frac{3}{2}}) : \frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}}

= \frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{6}{7} x^{2} (x - 1)^{\frac{3}{2}} + \frac{24}{35} x (x - 1)^{\frac{3}{2}} + \frac{328}{105} (x - 1)^{\frac{3}{2}} + C

What is the integral of (3x^2+4)sqrt(x-1) ?
The integral of (3x^2+4)sqrt(x-1) is 6/7 x^2(x-1)^{3/2}+24/35 x(x-1)^{3/2}+328/105 (x-1)^{3/2}+C