# Thread: Integral problem (pls help)

1. ## Integral problem (pls help)

Does anyone know how to solve this? I can't find similar problems anywhere

2. ## Re: Integral problem (pls help)

By definition of the Sine Integral:

$\displaystyle \dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt$

By the Fundamental Theorem of Calculus (sometimes called Part I):

$\displaystyle \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt = \dfrac{\sin(x^6)}{x^6}$

This is simply a direct application of the theorem. There is no calculation necessary. Just apply the Fundamental Theorem of Calculus and the answer is immediately apparent.

3. ## Re: Integral problem (pls help)

Originally Posted by SlipEternal
By definition of the Sine Integral:

$\displaystyle \dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt$

By the Fundamental Theorem of Calculus (sometimes called Part I):

$\displaystyle \dfrac{d}{dx}\int_0^x \dfrac{\sin \left(t^6\right)}{t^6}dt = \dfrac{\sin(x^6)}{x^6}$

This is simply a direct application of the theorem. There is no calculation necessary. Just apply the Fundamental Theorem of Calculus and the answer is immediately apparent.
I typed the exact same answer, but it was marked as incorrect. Thank you though!

4. ## Re: Integral problem (pls help)

Ok, I am not familiar with the Si function. I looked it up, but apparently do not understand it. According to Wolframalpha,

$\dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{6\sin(x^6)}{x}$

But, I have no clue how.

5. ## Re: Integral problem (pls help)

Oh, I just misread the description:

$\displaystyle \text{Si}(x^6) = \int_0^{x^6}\dfrac{\sin t}{t} dt$

So, by the Chain Rule, we have:

$\dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{d}{d(x^6)}\left[\text{Si}(x^6)\right]\dfrac{d(x^6)}{dx} = \dfrac{\sin(x^6)}{x^{\cancel{6}}}(6\cancel{x^5}) = \dfrac{6\sin(x^6)}{x}$

6. ## Re: Integral problem (pls help)

Originally Posted by SlipEternal
Ok, I am not familiar with the Si function. I looked it up, but apparently do not understand it. According to Wolframalpha,

$\dfrac{d}{dx}\left[\text{Si}(x^6)\right] = \dfrac{6\sin(x^6)}{x}$

But, I have no clue how.
thanks a lot!