What is the integral of 4sqrt(x)(x+10) ?

The integral of 4sqrt(x)(x+10) is 4(2/5 x^{5/2}+20/3 x^{3/2})+C

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integral of 4sqrt(x)(x+10)

\int 4 x (x + 10) d x

4 (\frac{2}{5} x^{\frac{5}{2}} + \frac{20}{3} x^{\frac{3}{2}}) + C

Solution steps

\int 4 x (x + 10) d x

Take the constant out:

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

= 4 \cdot \int x (x + 10) d x

Expand

x (x + 10) : x^{\frac{3}{2}} + 10 x

= 4 \cdot \int x^{\frac{3}{2}} + 10 x 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^{\frac{3}{2}} d x + \int 10 x d x)

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

\int 10 x d x = \frac{20}{3} x^{\frac{3}{2}}

= 4 (\frac{2}{5} x^{\frac{5}{2}} + \frac{20}{3} x^{\frac{3}{2}})

Add a constant to the solution

= 4 (\frac{2}{5} x^{\frac{5}{2}} + \frac{20}{3} x^{\frac{3}{2}}) + C

What is the integral of 4sqrt(x)(x+10) ?
The integral of 4sqrt(x)(x+10) is 4(2/5 x^{5/2}+20/3 x^{3/2})+C