What is the integral of (x^2+5)(sqrt(x)-5) ?

The integral of (x^2+5)(sqrt(x)-5) is 2/7 x^{7/2}-(5x^3)/3+10/3 x^{3/2}-25x+C

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integral of (x^2+5)(sqrt(x)-5)

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

\frac{2}{7} x^{\frac{7}{2}} - \frac{5 x ^{3}}{3} + \frac{10}{3} x^{\frac{3}{2}} - 25 x + C

Solution steps

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

Expand

(x^{2} + 5) (x - 5) : x^{\frac{5}{2}} - 5 x^{2} + 5 x - 25

= \int x^{\frac{5}{2}} - 5 x^{2} + 5 x - 25 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{5}{2}} d x - \int 5 x^{2} d x + \int 5 x d x - \int 25 d x

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

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

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

\int 25 d x = 25 x

= \frac{2}{7} x^{\frac{7}{2}} - \frac{5 x ^{3}}{3} + \frac{10}{3} x^{\frac{3}{2}} - 25 x

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} - \frac{5 x ^{3}}{3} + \frac{10}{3} x^{\frac{3}{2}} - 25 x + C

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What is the integral of (x^2+5)(sqrt(x)-5) ?
The integral of (x^2+5)(sqrt(x)-5) is 2/7 x^{7/2}-(5x^3)/3+10/3 x^{3/2}-25x+C