What is the integral of x*(2x^2+1)^{3/2} ?

The integral of x*(2x^2+1)^{3/2} is 1/5 x^2(2x^2+1)^{3/2}+1/10 (2x^2+1)^{3/2}+C

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integral of x*(2x^2+1)^{3/2}

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

\frac{1}{5} x^{2} (2 x^{2} + 1)^{\frac{3}{2}} + \frac{1}{10} (2 x^{2} + 1)^{\frac{3}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int 3 x^{3} (2 x^{2} + 1)^{\frac{1}{2}} d x = \frac{1}{10} (3 x^{2} - 1) (2 x^{2} + 1)^{\frac{3}{2}}

= \frac{1}{2} x^{2} (2 x^{2} + 1)^{\frac{3}{2}} - \frac{1}{10} (3 x^{2} - 1) (2 x^{2} + 1)^{\frac{3}{2}}

Simplify

\frac{1}{2} x^{2} (2 x^{2} + 1)^{\frac{3}{2}} - \frac{1}{10} (3 x^{2} - 1) (2 x^{2} + 1)^{\frac{3}{2}} : \frac{1}{5} x^{2} (2 x^{2} + 1)^{\frac{3}{2}} + \frac{1}{10} (2 x^{2} + 1)^{\frac{3}{2}}

= \frac{1}{5} x^{2} (2 x^{2} + 1)^{\frac{3}{2}} + \frac{1}{10} (2 x^{2} + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{5} x^{2} (2 x^{2} + 1)^{\frac{3}{2}} + \frac{1}{10} (2 x^{2} + 1)^{\frac{3}{2}} + C

\frac{d y}{d x} - \frac{6}{x} y = \frac{y ^{4}}{x ^{15}}, y (1) = 1

x \frac{d y}{d x} = - 4 y + x^{3} - x

\frac{d x}{d t} = 4 x (9 - x)

(3 x - 2)^{3} (x^{3} + 1)^{2}

\int x^{3} x^{2} + 12 d x

What is the integral of x*(2x^2+1)^{3/2} ?
The integral of x*(2x^2+1)^{3/2} is 1/5 x^2(2x^2+1)^{3/2}+1/10 (2x^2+1)^{3/2}+C