What is the integral of (4+x^2)(9-sqrt(x)) ?

The integral of (4+x^2)(9-sqrt(x)) is 36x-8/3 x^{3/2}+3x^3-2/7 x^{7/2}+C

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integral of (4+x^2)(9-sqrt(x))

\int (4 + x^{2}) (9 - x) d x

36 x - \frac{8}{3} x^{\frac{3}{2}} + 3 x^{3} - \frac{2}{7} x^{\frac{7}{2}} + C

Solution steps

\int (4 + x^{2}) (9 - x) d x

Expand

(4 + x^{2}) (9 - x) : 36 - 4 x + 9 x^{2} - x^{\frac{5}{2}}

= \int 36 - 4 x + 9 x^{2} - x^{\frac{5}{2}} 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 36 d x - \int 4 x d x + \int 9 x^{2} d x - \int x^{\frac{5}{2}} d x

\int 36 d x = 36 x

\int 4 x d x = \frac{8}{3} x^{\frac{3}{2}}

\int 9 x^{2} d x = 3 x^{3}

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

= 36 x - \frac{8}{3} x^{\frac{3}{2}} + 3 x^{3} - \frac{2}{7} x^{\frac{7}{2}}

Add a constant to the solution

= 36 x - \frac{8}{3} x^{\frac{3}{2}} + 3 x^{3} - \frac{2}{7} x^{\frac{7}{2}} + C

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What is the integral of (4+x^2)(9-sqrt(x)) ?
The integral of (4+x^2)(9-sqrt(x)) is 36x-8/3 x^{3/2}+3x^3-2/7 x^{7/2}+C