What is the integral of x(3x^2+x^{1/2}) ?

The integral of x(3x^2+x^{1/2}) is (3x^4)/4+2/5 x^{5/2}+C

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integral of x(3x^2+x^{1/2})

\int x (3 x^{2} + x^{\frac{1}{2}}) d x

\frac{3 x ^{4}}{4} + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

\int x (3 x^{2} + x^{\frac{1}{2}}) d x

Expand

x (3 x^{2} + x^{\frac{1}{2}}) : 3 x^{3} + x^{\frac{3}{2}}

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

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

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

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

Add a constant to the solution

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

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