What is the integral of x^5sqrt(x^3)+4 ?

The integral of x^5sqrt(x^3)+4 is 2/15 x^{15/2}+4x+C

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integral of x^5sqrt(x^3)+4

\int x^{5} x^{3} + 4 d x

\frac{2}{15} x^{\frac{15}{2}} + 4 x + C

Solution steps

\int x^{5} x^{3} + 4 d x

x^{3} = x^{\frac{3}{2}},

assuming

x \geq 0

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

\int x^{5} x^{\frac{3}{2}} d x = \frac{2}{15} x^{\frac{15}{2}}

\int 4 d x = 4 x

= \frac{2}{15} x^{\frac{15}{2}} + 4 x

Add a constant to the solution

= \frac{2}{15} x^{\frac{15}{2}} + 4 x + C

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16 y^{^{''}} + 40 y^{^{'}} + 25 y = 0

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(6 x + 1) y^{2} (\frac{d y}{d x}) + 3 x^{2} + 2 y^{3} = 0

What is the integral of x^5sqrt(x^3)+4 ?
The integral of x^5sqrt(x^3)+4 is 2/15 x^{15/2}+4x+C