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

The integral of sqrt(x)(3x+1) is 2/3 x^{3/2}+6/5 x^{5/2}+C

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integral of sqrt(x)(3x+1)

\int x (3 x + 1) d x

\frac{2}{3} x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}} + C

Solution steps

\int x (3 x + 1) d x

Apply Integration By Parts

: 2 x^{\frac{5}{2}} + \frac{2}{3} x^{\frac{3}{2}} - \int 2 x^{\frac{3}{2}} d x

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

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

= 2 x^{\frac{5}{2}} + \frac{2}{3} x^{\frac{3}{2}} - \frac{4}{5} x^{\frac{5}{2}}

Simplify

2 x^{\frac{5}{2}} + \frac{2}{3} x^{\frac{3}{2}} - \frac{4}{5} x^{\frac{5}{2}} : \frac{2}{3} x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}}

= \frac{2}{3} x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + \frac{6}{5} x^{\frac{5}{2}} + C

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What is the integral of sqrt(x)(3x+1) ?
The integral of sqrt(x)(3x+1) is 2/3 x^{3/2}+6/5 x^{5/2}+C