What is the integral of 4x(1+sqrt(x)) ?

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

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integral of 4x(1+sqrt(x))

\int 4 x (1 + x) d x

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

Solution steps

\int 4 x (1 + x) d x

Take the constant out:

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

= 4 \cdot \int x (1 + x) d x

Expand

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

= 4 \cdot \int x + 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

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

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

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

= 4 (\frac{x ^{2}}{2} + \frac{2}{5} x^{\frac{5}{2}})

Add a constant to the solution

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

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