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

The integral of x(sqrt(x)+x)^2 is (x^3)/2+4/5 x^{7/2}+(x^4)/3-(x^3)/6-8/35 x^{7/2}-(x^4)/(12)+C

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

\int x (x + x)^{2} d x

\frac{x ^{3}}{2} + \frac{4}{5} x^{\frac{7}{2}} + \frac{x ^{4}}{3} - \frac{x ^{3}}{6} - \frac{8}{35} x^{\frac{7}{2}} - \frac{x ^{4}}{12} + C

Solution steps

\int x (x + x)^{2} d x

Apply Integration By Parts

: x (\frac{x ^{2}}{2} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3}) - \int \frac{x ^{2}}{2} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3} d x

= x (\frac{x ^{2}}{2} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3}) - \int \frac{x ^{2}}{2} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3} d x

\int \frac{x ^{2}}{2} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3} d x = \frac{x ^{3}}{6} + \frac{8}{35} x^{\frac{7}{2}} + \frac{x ^{4}}{12}

= x (\frac{x ^{2}}{2} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3}) - (\frac{x ^{3}}{6} + \frac{8}{35} x^{\frac{7}{2}} + \frac{x ^{4}}{12})

Simplify

x (\frac{x ^{2}}{2} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3}) - (\frac{x ^{3}}{6} + \frac{8}{35} x^{\frac{7}{2}} + \frac{x ^{4}}{12}) : \frac{x ^{3}}{2} + \frac{4}{5} x^{\frac{7}{2}} + \frac{x ^{4}}{3} - \frac{x ^{3}}{6} - \frac{8}{35} x^{\frac{7}{2}} - \frac{x ^{4}}{12}

= \frac{x ^{3}}{2} + \frac{4}{5} x^{\frac{7}{2}} + \frac{x ^{4}}{3} - \frac{x ^{3}}{6} - \frac{8}{35} x^{\frac{7}{2}} - \frac{x ^{4}}{12}

Add a constant to the solution

= \frac{x ^{3}}{2} + \frac{4}{5} x^{\frac{7}{2}} + \frac{x ^{4}}{3} - \frac{x ^{3}}{6} - \frac{8}{35} x^{\frac{7}{2}} - \frac{x ^{4}}{12} + C

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What is the integral of x(sqrt(x)+x)^2 ?
The integral of x(sqrt(x)+x)^2 is (x^3)/2+4/5 x^{7/2}+(x^4)/3-(x^3)/6-8/35 x^{7/2}-(x^4)/(12)+C