What is the integral of f(x)=sqrt(x)(6+5x) ?

The integral of f(x)=sqrt(x)(6+5x) is 4x^{3/2}+2x^{5/2}+C

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integral of f(x)=sqrt(x)(6+5x)

integral

f (x) = x (6 + 5 x)

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

Solution steps

\int x (6 + 5 x) d x

Expand

x (6 + 5 x) : 6 x + 5 x^{\frac{3}{2}}

= \int 6 x + 5 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 6 x d x + \int 5 x^{\frac{3}{2}} d x

\int 6 x d x = 4 x^{\frac{3}{2}}

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

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

Add a constant to the solution

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

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What is the integral of f(x)=sqrt(x)(6+5x) ?
The integral of f(x)=sqrt(x)(6+5x) is 4x^{3/2}+2x^{5/2}+C