What is the integral of 2x(sqrt(x)+x) ?

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

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integral of 2x(sqrt(x)+x)

\int 2 x (x + x) d x

2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3}) + C

Solution steps

\int 2 x (x + x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x (x + x) d x

Expand

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

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

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

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

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

= 2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{x ^{3}}{3})

Add a constant to the solution

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

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