Question

Question #633a8

Answer 1

`intydx=-2cost`

Explanation:

Giben:
`y=sinx^(1/2)/x^(1/2)`
Integrate the above function with respect to x
`intydx=intsinx^(1/2)/x^(1/2)dx`

Substitute `x^(1/2)` be t
Differentiating both sides with respect to t, we obtain
We know that `d/dx(x^n) = nx^(n-1)`
When t = `x^(1/2)`, n = 1/2 and n-1 = 1/2-1 = -1/2
Thus, `dt/dx = 1/2x^(-1/2)`
`dt/dx=1/(2x^(1/2))`
Separating variables t and x, we have
`dt = dx/(2x^(1/2))`
implies
`dx/(2x^(1/2))=dt`
or
`dx/x^(1/2)=2dt`
Substituting for `x^(1/2)` and `dx/x^(1/2)` in
`intydx=intsinx^(1/2)/x^(1/2)dx=int(sinx^(1/2))(1/(x^(1/2))dx)`, we have
`intydx=intsint(2dt)`
Simplifying
`intydx=2intsintdt`
`intydx=-2cost`