What is the integral of sqrt(x)(x^6+x-5) ?

The integral of sqrt(x)(x^6+x-5) is 2/15 x^{15/2}+2/5 x^{5/2}-10/3 x^{3/2}+C

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integral of sqrt(x)(x^6+x-5)

\int x (x^{6} + x - 5) d x

\frac{2}{15} x^{\frac{15}{2}} + \frac{2}{5} x^{\frac{5}{2}} - \frac{10}{3} x^{\frac{3}{2}} + C

Solution steps

\int x (x^{6} + x - 5) d x

Expand

x (x^{6} + x - 5) : x^{\frac{13}{2}} + x^{\frac{3}{2}} - 5 x

= \int x^{\frac{13}{2}} + x^{\frac{3}{2}} - 5 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{\frac{13}{2}} d x + \int x^{\frac{3}{2}} d x - \int 5 x d x

\int x^{\frac{13}{2}} d x = \frac{2}{15} x^{\frac{15}{2}}

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

\int 5 x d x = \frac{10}{3} x^{\frac{3}{2}}

= \frac{2}{15} x^{\frac{15}{2}} + \frac{2}{5} x^{\frac{5}{2}} - \frac{10}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{15} x^{\frac{15}{2}} + \frac{2}{5} x^{\frac{5}{2}} - \frac{10}{3} x^{\frac{3}{2}} + C

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What is the integral of sqrt(x)(x^6+x-5) ?
The integral of sqrt(x)(x^6+x-5) is 2/15 x^{15/2}+2/5 x^{5/2}-10/3 x^{3/2}+C