What is the integral of x(1+x^2)^{5/2} ?

The integral of x(1+x^2)^{5/2} is 1/7 x^2(1+x^2)^{5/2}+1/7 (x^2+1)^{5/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of x(1+x^2)^{5/2}

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

\frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}} + C

Solution steps

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

Apply Integration By Parts

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

\int \frac{5}{2} x^{3} (1 + x^{2})^{\frac{3}{2}} d x = \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}}

= \frac{1}{2} x^{2} (1 + x^{2})^{\frac{5}{2}} - \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}}

Simplify

\frac{1}{2} x^{2} (1 + x^{2})^{\frac{5}{2}} - \frac{1}{14} (5 x^{2} - 2) (x^{2} + 1)^{\frac{5}{2}} : \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}

= \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}

Add a constant to the solution

= \frac{1}{7} x^{2} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}} + C

\int 24 csc (4 x) sec (4 x) d x

\int \frac{1}{( x - 4 ) ^{4}} d x

\int e^{2 x + 4} d x

\int - \frac{y}{2} d y

\int \frac{1}{9 x ^{2} - 12 x - 5} d x

What is the integral of x(1+x^2)^{5/2} ?
The integral of x(1+x^2)^{5/2} is 1/7 x^2(1+x^2)^{5/2}+1/7 (x^2+1)^{5/2}+C