Question

What is `f(x) = int xsqrt(5x-2) dx` if `f(3) = 3`?

Answer 1

`f(x) = 1/25 (5x-2)^(3/2)( 2x + 8/15 ) + 3-(1274 sqrt(13))/375`

Explanation:

`f(x) = int dx qquad xsqrt(5x-2)`

you could use IBP but a sub should trivialise this

so `z = 5x-2, dz = 5 dx`

`f(z) = 1/5 int dz qquad (z+2)/5 sqrt(z)`

`= 1/25 int dz qquad z^(3/2)+2 sqrt(z)`

`= 1/25 ( 2/5z^(5/2)+2*2/3 z^(3/2) ) + C`

`= 1/25 ( 2/5z^(5/2)+4/3 z^(3/2) ) + C`

`= 1/25z^(3/2) ( 2/5z+4/3 ) + C`

`f(x) = 1/25 (5x-2)^(3/2)( 2/5(5x-2)+4/3 ) + C`

`f(x) = 1/25 (5x-2)^(3/2)( 2x + 8/15 ) + C`

`f( 3) = 3 implies C = 3-(1274 sqrt(13))/375`