What is the integral of ((x^{1/2}+8)^4)/(x^{1/2)} ?

The integral of ((x^{1/2}+8)^4)/(x^{1/2)} is 2/5 x^{5/2}+16x^2+256x^{3/2}+2048x+8192x^{1/2}+C

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integral of ((x^{1/2}+8)^4)/(x^{1/2)}

\int \frac{( x ^{\frac{1}{2}} + 8 ) ^{4}}{x ^{\frac{1}{2}}} d x

\frac{2}{5} x^{\frac{5}{2}} + 16 x^{2} + 256 x^{\frac{3}{2}} + 2048 x + 8192 x^{\frac{1}{2}} + C

Solution steps

\int \frac{( x ^{\frac{1}{2}} + 8 ) ^{4}}{x ^{\frac{1}{2}}} d x

Long division

\frac{( x ^{\frac{1}{2}} + 8 ) ^{4}}{x ^{\frac{1}{2}}} : x^{\frac{3}{2}} + 32 x + 384 x^{\frac{1}{2}} + 2048 + \frac{4096}{x ^{\frac{1}{2}}}

= \int x^{\frac{3}{2}} + 32 x + 384 x^{\frac{1}{2}} + 2048 + \frac{4096}{x ^{\frac{1}{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 x^{\frac{3}{2}} d x + \int 32 x d x + \int 384 x^{\frac{1}{2}} d x + \int 2048 d x + \int \frac{4096}{x ^{\frac{1}{2}}} d x

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

\int 32 x d x = 16 x^{2}

\int 384 x^{\frac{1}{2}} d x = 256 x^{\frac{3}{2}}

\int 2048 d x = 2048 x

\int \frac{4096}{x ^{\frac{1}{2}}} d x = 8192 x^{\frac{1}{2}}

= \frac{2}{5} x^{\frac{5}{2}} + 16 x^{2} + 256 x^{\frac{3}{2}} + 2048 x + 8192 x^{\frac{1}{2}}

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} + 16 x^{2} + 256 x^{\frac{3}{2}} + 2048 x + 8192 x^{\frac{1}{2}} + C

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What is the integral of ((x^{1/2}+8)^4)/(x^{1/2)} ?
The integral of ((x^{1/2}+8)^4)/(x^{1/2)} is 2/5 x^{5/2}+16x^2+256x^{3/2}+2048x+8192x^{1/2}+C