Question

Find f. f'(x)=(sqrt(x))(6+10x) f(1)=11

Answer 1

It is a matter of integrating f′:

$f(x)=\int \sqrt{x}(6+10x)dx, \\ \\f(x)=\int 6x^{1/2}dx+\int10x^{3/2}dx, \\ \\f(x)=4x^{3/2}(x+1)+C.$

Combine the constants. Then use the initial condition to find C:

$11=8+C\implies C=3$

Therefore,

$f(x)=4x^{3/2}(x+1)+3.$