YAMathT: Need a small understanding for antiderivatives...

[Ricemarine]

f(x) = x^1/3 (or the third root whatever, don't know terminology) + (5 / x^6)...

So when I found the anti derivative...

F(x) = (x^4/3)/(4/3) + (5x^-5 + C).

F(x) = 3/4 x ^(4/3) - x^-5 + C...

The Answer should be.
F(x) = 3/4 x^(3/4) - x^(-5) + C(sub1) if x < 0.
F(x) = 3/4 x^(3/4) - x^(-5) + C(sub2) if x > 0.


How did they come to this resolution?. Seriously, integrals are a PITA...
Thanks for your help if you do help .

 

[HaxorNubcake]

Um I get

(3/4)x^(4/3)-x^(-5)+Cx (same answer)

So...I conclude that the provided solution is wrong

 

[HaxorNubcake]

http://integrals.wolfram.com/index.jsp

Mathematica agrees as well

 

[HaxorNubcake]

http://pics.bbzzdd.com/users/HaxorNubcake/integral.JPG

 

[Ricemarine]

Originally posted by: HaxorNubcake
http://pics.bbzzdd.com/users/HaxorNubcake/integral.JPG

Hmm... typo in the book I guess... Thx.

 

[randumb]

Originally posted by: Ricemarine

Originally posted by: HaxorNubcake
http://pics.bbzzdd.com/users/HaxorNubcake/integral.JPG

Hmm... typo in the book I guess... Thx.

No. The book is right. The reason you have two different constants is because of the discontinuity at 0.

 

[Ricemarine]

Originally posted by: randumb

Originally posted by: Ricemarine

Originally posted by: HaxorNubcake
http://pics.bbzzdd.com/users/HaxorNubcake/integral.JPG

Hmm... typo in the book I guess... Thx.

No. The book is right. The reason you have two different constants is because of the discontinuity at 0.

But why is it 3/4 x^3/4? and not 3/4 x^4/3.

 

[randumb]

Originally posted by: Ricemarine

Originally posted by: randumb

Originally posted by: Ricemarine

Originally posted by: HaxorNubcake
http://pics.bbzzdd.com/users/HaxorNubcake/integral.JPG

Hmm... typo in the book I guess... Thx.

No. The book is right. The reason you have two different constants is because of the discontinuity at 0.

But why is it 3/4 x^3/4? and not 3/4 x^4/3.

Oh. Sorry, that is a typo. I skimmed over and assumed it was the same.

But there should be two constants.

 

[otispunkmeyer]

why on earth do you call it the anti-derivative.

whats wrong with the word integral.

also im a bit rusty on integration, but why is it Cx?

wouldnt it just be C? since there was no C in the starting equation, and you add a constant to any integral thats done without limits

or is it just the way you guys write it.

EDIT: oh hang on, you had a C in the one you sent to that mathematica place, but the OP has no C in his, unless he edited out. sorry lol!

 

[Ricemarine]

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.

whats wrong with the word integral.

also im a bit rusty on integration, but why is it Cx?

wouldnt it just be C? since there was no C in the starting equation, and you add a constant to any integral thats done without limits

or is it just the way you guys write it.

EDIT: oh hang on, you had a C in the one you sent to that mathematica place, but the OP has no C in his, unless he edited out. sorry lol!

Actually. I didn't edit mine, he just added an extra C in the original formula... But um yeah... I guess the book made a mistake if no one else can solve it

 

[jman19]

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.

whats wrong with the word integral.

Anti-derivative is just another name for an indefinate integral.

 

[Safeway]

Originally posted by: jman19

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.
whats wrong with the word integral.

Anti-derivative is just another name for an indefinate integral.

QFT. What's wrong with calling something an integral?

 

[jman19]

Originally posted by: Safeway

Originally posted by: jman19

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.
whats wrong with the word integral.

Anti-derivative is just another name for an indefinate integral.

QFT. What's wrong with calling something an integral?

For students who began their calculus education with derivatives, calling indefinite integrals anti-derivatives is perhaps an easier way to understand the process of integration.

 

[VTHodge]

Originally posted by: Safeway

Originally posted by: jman19

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.
whats wrong with the word integral.

Anti-derivative is just another name for an indefinate integral.

QFT. What's wrong with calling something an integral?

The way I remember it is that Newton used one term and Leibniz used the other. Of course now I can't find any reference to back that up.

Also, your exponent is correct, but the piecewise answer is important to cover the discontinuity at zero (I know I'm just agreeing with others here).

 

[Safeway]

Originally posted by: VTHodge

Originally posted by: Safeway

Originally posted by: jman19

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.
whats wrong with the word integral.

Anti-derivative is just another name for an indefinate integral.

QFT. What's wrong with calling something an integral?

The way I remember it is that Newton used one term and Leibniz used the other. Of course now I can't find any reference to back that up.

From middle school to graduate school, the only term used in my education has been integral. When we were first taught about integrals, the term anti-derivative was used once to get the point across.

 

[Matthias99]

Originally posted by: Safeway
From middle school to graduate school, the only term used in my education has been integral. When we were first taught about integrals, the term anti-derivative was used once to get the point across.

I got both sort of interchangeably from different sources.

As for reasoning: It's shorter to say 'anti-derivative' than 'indefinite integral'? And just using 'integral' is less specific, since you're not specifying whether it's over the entire function range or just part of it.

 

[HaxorNubcake]

Originally posted by: Ricemarine

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.

whats wrong with the word integral.

also im a bit rusty on integration, but why is it Cx?

wouldnt it just be C? since there was no C in the starting equation, and you add a constant to any integral thats done without limits

or is it just the way you guys write it.

EDIT: oh hang on, you had a C in the one you sent to that mathematica place, but the OP has no C in his, unless he edited out. sorry lol!

Actually. I didn't edit mine, he just added an extra C in the original formula... But um yeah... I guess the book made a mistake if no one else can solve it

Oops Oh well, as long as you figured it out

 

[Ricemarine]

Originally posted by: HaxorNubcake

Originally posted by: Ricemarine

Originally posted by: otispunkmeyer
why on earth do you call it the anti-derivative.

whats wrong with the word integral.

also im a bit rusty on integration, but why is it Cx?

wouldnt it just be C? since there was no C in the starting equation, and you add a constant to any integral thats done without limits

or is it just the way you guys write it.

EDIT: oh hang on, you had a C in the one you sent to that mathematica place, but the OP has no C in his, unless he edited out. sorry lol!

Actually. I didn't edit mine, he just added an extra C in the original formula... But um yeah... I guess the book made a mistake if no one else can solve it

Oops Oh well, as long as you figured it out

Yeap, thanks for your help haxornubcake .

 

[silverpig]

I always use anti-derivative to mean the function, and integral to mean a value.

So I guess I use anti-derivative - indefinite integral interchangably, and integral - definite integral interchangably.