What is the integral of (7x^3+x^2)/(sqrt(x)) ?

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

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integral of (7x^3+x^2)/(sqrt(x))

\int \frac{7 x ^{3} + x ^{2}}{x} d x

2 x^{\frac{7}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

\int \frac{7 x ^{3} + x ^{2}}{x} d x

Expand

\frac{7 x ^{3} + x ^{2}}{x} : \frac{7 x ^{3}}{x} + \frac{x ^{2}}{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 \frac{7 x ^{3}}{x} d x + \int \frac{x ^{2}}{x} d x

\int \frac{7 x ^{3}}{x} d x = 2 x^{\frac{7}{2}}

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

= 2 x^{\frac{7}{2}} + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 2 x^{\frac{7}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

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