What is the Antiderivative of (f(x) = √(x^5 ) - 4/√(5&x))?

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1. This antiderivative question asks: Find f.
The equation reads: f prime of x = square root of x to the 5th power minus 4 divided by the fifth root of x. (see below)

f´ (x) = √(x^5 ) - 4/√(5&x)

The answer in the back of the book is f(x) = (2/7)X^(7/2) - 5X^(4/5) + C but I got stuck trying to work it out.


2. Here are the relevant equations:
If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is: F(x) + C

Antidifferentiation formulas:

Function = cf(x) Particular antiderivative = cF(x)
Function = f(x) + g(x) Particular antiderivative = F(x) + G(x)
Function = x^n (n not equal -1) Particular antiderivative (x^n+1)/n+1


3. My attempt at a solution reads as follows:
X to the fifth power raised to the half power minus 4 times 5X raised to the minus a half

= 〖〖(X〗^(5))〗^(1/(2 ))-4(5X^((-1)/2))

 

[Dick]

The fifth root of x in terms of power is (x)^(1/5). So you have (x^5)^(1/2)-4/(x)^(1/5). Do you remember your rules of exponents? Can you change that into the sum of two simple powers of x?

 

[JOhnJDC]

EDIT: Removed at Dick's request.

 

[01010011]

The fifth root of x in terms of power is (x)^(1/5). So you have (x^5)^(1/2)-4/(x)^(1/5). Do you remember your rules of exponents? Can you change that into the sum of two simple powers of x?

EDIT: Removed at Dick's request.

Thank you for your replies Dick and JohnJDC.

Ok, I think I remember the rules of exponents, so here goes another attempt:

Question: f prime x = sq.root of x to the 5th power - 4 divided by the 5th root of x

Simplify first:

= ((x^5)^ 1/2) - 4/ (x^1/5)

= (x^5/2) - 4x^-1/5

most general antiderivative:

= [x^(5/2 + 1) divided by (5/2 + 1)] - [4x^(-1/5 + 1) divided by (-1/5 + 1) ]

= [ (x ^ 7/2) divided by (7/2) ] - [ (4x^4/5) divided by (4/5) ] + C

= (2/7x^7/2) - [(4x^4/5) divided by 4/5 ] + C

= (2/7x^7/2) - (5/4 * 4x^4/5) + C

= (2/7x^7/2) - (5x^4/5) + C

 

[JOhnJDC]

most general antiderivative:

= [x^(5/2 + 1) divided by (5/2 + 1)] - [4x^(-1/5 + 1) divided by (-1/5 + 1) ]

= [ (x ^ 7/2) divided by (7/2) ] - [ (4x^4/5) divided by (4/5) ] + C

Answer I got is = (2/7x^7/2) - [(4x^4/5) divided by 4/5 ] + C

However, the answer in the book is f(x) = (2/7)X^(7/2) -5X^(4/5) + C Where did I go wrong

You're there--you just need to simplify the term on the right. Consider this:
3x/(3/5) = 3x(5/3) = 5x

 

[01010011]

Thank you JOhnJDC for your help

 

[Dick]

EDIT: Removed at Dick's request.

Thanks!