Question

What is `f(x) = int xsqrt(3x) dx` if `f(3) = 0`?

Answer 1

`=>f(x) = (2sqrt(3))/5x^(5/2) -54/5`

Explanation:

`f(x) = int xsqrt(3x) dx`
`=>f(x) = sqrt(3)int xsqrt(x) dx`
`=>f(x) = sqrt(3)int x^(3/2) dx`
`=>f(x) = sqrt(3)x^(3/2+1)/(3/2+1) +c` [where c = Integration constant]

`=>f(x) = (2sqrt(3))/5x^(5/2) +c.....(1)`

Again given condition is `f(3)=0`
So
`=>f(3) = (2sqrt(3))/5xx3^(5/2) +c`
`=>0= 2/5xx3^(5/2+1/2) +c`
`=>0= 2/5xx3^3 +c`
`=>0= 54/5 +c`
`=>c= -54/5`
Hence we have , substituting the value of c in eq(1)

`=>f(x) = (2sqrt(3))/5x^(5/2) -54/5`