Quick explanation to an integration question

[TeaStorm]

Hello! I have a quick question that confused me for some time.

I have this question:
Determine[MATH]\frac{d}{dx}\int\limits_{x^2}^0 \frac{1}{t^4+1}dt[/MATH]
My first idea was to solve it through this method:
1. [MATH]=\frac{d}{dx}\int\limits_0^{x^2} \frac{-1}{t^4+1}dt[/MATH]2. [MATH]= \frac{-1}{(x^2)^4+1}[/MATH] (Fundamental Theorem)

Though this turned out to be wrong, and the correct answer turned out to be [MATH]-\frac{1}{x^8+1}\cdot2x[/MATH]
Can someone please explain to me why my method turned out to be wrong?

 

[tkhunny]

You overlooked the Chain Rule when you manufactured your derivative.

 

[TeaStorm]

You overlooked the Chain Rule when you manufactured your derivative.

Thank you for the quick reply.

Does this mean I could use a formula like this: [MATH]\int\limits_a^uf(t) dt=f(u)\cdot{u'}[/MATH]?
Or does this Chain Rule only apply in specific cases?

 

[Steven G]

There is a theorem that states that [math]\dfrac{d}{dx}\int _{h(x)}^{j(x)}f(t)dt = f(j(x))j'(x) - f(h(x))h'(x)[/math]
There is no need to compute [math]\int f(x)dx!!![/math]

 

[TeaStorm]

There is a theorem that states that [math]\dfrac{d}{dx}\int _{h(x)}^{j(x)}f(t)dt = f(j(x))j'(x) - f(h(x))h'(x)[/math]
There is no need to compute [math]\int f(x)dx!!![/math]

Thank you very much for the response, I'll make good use of this theorem ?
Thread is now marked as solved.

 

[Steven G]

Thank you for the quick reply.

Does this mean I could use a formula like this: [MATH]\int\limits_a^uf(t) dt=f(u)\cdot{u'}[/MATH]?

No, you need to do the same for the lower limit.

If by chance you are saying that u is a function of x and a is a constant, then what you stated is correct. My question to you is if you fail to identify YOUR variables then how is anyone to know what you are thinking when it comes to your variables?

 

[TeaStorm]

No, you need to do the same for the lower limit.

If by chance you are saying that u is a function of x and a is a constant, then what you stated is correct. My question to you is if you fail to identify YOUR variables then how is anyone to know what you are thinking when it comes to your variables?

This is true. I intended to identify a as a constant and not a function, and I as a function of x. I should have specified this more clearly.