Evaluate: `int(x^(1//2))/(1+x^(3//4))\ dx`

Answer

N/a

Explanation

Let `int = int(x^(1//2))/(1+x^(3//4))dx`
Put `x=t^(4) rArr dx = 4t^(3)dt`
`:. I = 4int(t^(2)(t^(3)))/(1+t)dt = 4int(t^(2)- (t^(2))/(1+t^(3)))dt`
`I = 4intt^(2)dt - 4int(t^(2))/(1+t^(3))dt`
`I = I_(1) - I_(2)`
`I_(1) = 4intt^(2)dt = 4.(t^(3))/(3)+C_(1) = 4/3x^(3//4) + C_(1)`
Now, `I_(2)=4int(t^(2))/(1+t^(3))dt`
Again, put `1 + t^(3)= z rArr 3t^(2)dt = dz`
`rArr t^(2)dt = 1/3 dz = 4/3 int1/z dz`
` = 4/3 log|z|+C_(2)=4/3log|(1+t^(3))|+C_(2)`
` = 4/3 log|(1+x^(3//4))|+C_(2)`
`:. I = 4/3 x^(3//4) + C_(1) - 4/3log|(1+x^(3//4))|-C_(2)`
`= 4/3x^(3//4)-log|(1+x^(3//4))+C , [:' C = C_(1) + C_(2)]`