What is the integral of sqrt(x)^5 ?

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

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integral of sqrt(x)^5

\int x^{5} d x

\frac{2}{7} x^{\frac{7}{2}} + C

Solution steps

\int (x)^{5} d x

Apply Integration By Parts

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

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

\int \frac{5}{2} x^{\frac{5}{2}} d x = \frac{5}{7} x^{\frac{7}{2}}

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

Simplify

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

= \frac{2}{7} x^{\frac{7}{2}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} + C

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