1. ## limit and integration

Evaluate $\displaystyle {\lim }\limits_{x \to 0} \frac{\int_0^x\,e^{t^2}(e^t-1)^2\, dt}{x(1-cos x)}$

2. Originally Posted by Sambit
Evaluate $\displaystyle {\lim }\limits_{x \to 0} \frac{\int_0^x\,e^{t^2}(e^t-1)^2\, dt}{x(1-cos x)}$
Apply l'Hospital's Rule (and recall the Fundamental Theorem of Calculus when doing so).

3. Hi, I will continue mr fantastic idea...

Why you can apply l'Hospital's rule?

Let we see what happens to the whole expession when $\displaystyle x\to 0$:

$\displaystyle \frac{\int_0^0\,e^{t^2}(e^t-1)^2\, dt}{0(1-cos 0)}=\frac{0}{0}$

Next, $\displaystyle {x (1-cos(x))}' = {x sin(x)-cos(x)+1}\neq 0$ for every $\displaystyle x\neq0$

Applying l'Hospital's rule:
Now by the Fundamental Theorem of Calculus:

$\displaystyle {\int_0^x\,e^{t^2}(e^t-1)^2\ dt}}'=e^{x^2}(e^x-1)^2$

$\displaystyle {\lim }\limits_{x \to 0} {\frac{(\int_0^x\,e^{t^2}(e^t-1)^2\, dt)'}{(x(1-cos x))'}={\lim }\limits_{x \to 0} \frac{e^{x^2}(e^x-1)^2}{x sin(x)-cos(x)+1}$

The limit is still from the form of $\displaystyle \frac{0}{0}$, applying l'Hospital's rule again...

4. ...............in this manner, if L'Hospitals rule is applied, at the $\displaystyle 3rd$ oreder differentiation, we get:-
$\displaystyle \frac{((8x^3+24x^2+36x+20)e^{2x}+(-16x^3-24x^2-36x-14)e^x+8x^3+12x)e^{x^2}}{3cos x-xsin x}$

putting x=0, we get:- $\displaystyle 6/3 = 2$

am i right?

5. Originally Posted by Sambit
...............in this manner, if L'Hospitals rule is applied, at the $\displaystyle 3rd$ oreder differentiation, we get:-
$\displaystyle \frac{((8x^3+24x^2+36x+20)e^{2x}+(-16x^3-24x^2-36x-14)e^x+8x^3+12x)e^{x^2}}{3cos x-xsin x}$

putting x=0, we get:- $\displaystyle 6/3 = 2$

am i right?
No.

Here is the rest of post #3:

Originally Posted by Also sprach Zarathustra
$\displaystyle {\lim }\limits_{x \to 0} \frac{e^{x^2}(e^x-1)^2)'}{(x sin(x)-cos(x)+1)'}={\lim }\limits_{x \to 0} \frac{2xe^{x^2}(e^x-1)^2+2e^{x^2}e^x(e^x-1)}{2sin(x)+xcos(x)}$

... l'Hospital's rule again...

$\displaystyle {\lim }\limits_{x \to 0} \frac{(2xe^{x^2}(e^x-1)^2+2e^{x^2}e^x(e^x-1))'}{(2sin(x)+xcos(x))'}=2{\lim }\limits_{x \to 0} \frac{-(e^{x^2} (2 x^2+e^{2 x} (2 x^2+4 x+3)-e^x (4 x^2+4 x+3)+1)}{xsin(x)-3cos(x)}...=\frac{2}{3}$

6. oh yes... actually my denominator was right, but i differentiated the numerator 4 times instead of 3 times....now i have got it correct...

### integration cos x limit -a to a

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