Evaluate the following definite or indefinite integrals: |   (2/x+4)/sqrtx dx |

Expert Answers

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`(2/x+4)/sqrt(x)= 2/x^(3/2) + 4/sqrt(x)`

So

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

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

`= 2(1/(-1/2))x^(-1/2) + 4(1/(1/2))x^(1/2) + C`

` = -4x^(-1/2) + 8x^(1/2) + C`

`= -4/sqrt(x) + 8sqrt(x) + C = (-4+8x)/sqrt(x) + C`
 so

`int (2/x+4)/sqrt(x) dx = (8x-4)/sqrt(x) + C`

we can check with the quotient property taking the derivative of (8x-4)/sqrt(x)

` (d)/(dx) (8x-4)/sqrt(x) = ((sqrt(x)(8) - (8x-4)/(2sqrt(x))))/(sqrt(x))^2`

`= (8sqrt(x) - 4x/sqrt(x) + 2/sqrt(x))/x = (8x - 4x + 2)/(xsqrt(x)) = (4x+2)/(xsqrt(x)) = (4+2/x)/sqrt(x)`

So the answer is

`int (2/x+4)/sqrt(x) dx = (8x-4)/sqrt(x) + C`

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