Question

Question #a13e8

Answer 1

`color(blue)(-= ( 2sqrt(3) )/3 x + c_1`

Explanation:

The first thing to do is re-write...

`sqrt( (4x)/(3x) ) = sqrt( (4 cancel(x) )/ ( 3 cancel(x) ) ) = sqrt(4/3)`

`=> 2/(sqrt(3)) = (2*sqrt(3)) / ( sqrt(3) * sqrt(3) ) = (2sqrt(3))/3`

`int sqrt( (4x)/(3x) ) dx -= int ( 2sqrt(3) )/ 3 dx -= ( 2sqrt(3))/3 int 1 dx`

We can use the reverse power rule..

`int x^n dx = 1/(n+1) x^(n+1) + c`

`( 2sqrt(3) ) / 3 int 1 dx -= ( 2sqrt(3)) / 3 int x^0 dx`

`-= ( 2sqrt(3))/3 * ( 1/(0+1) x^(0+1) + c_0 )`

`-= ( 2sqrt(3) ) / 3 x + (2sqrt(3))/3 c_0`

`( 2sqrt(3) )/3 c_0` is just another constant `c_1`

`color(blue)(-= ( 2sqrt(3) )/3 x + c_1`