Definite Integral

I tried it this way........

I = ∫ (X/1! - x^3/3! + x^5/5! .....)/x dx

= ∫ (1- x^2/3! + x^4/5! - .....) dx

= (x - x^3/3.3! +x^5/5.5!....)

Applying limits now is troublesome????????

Assume : I(a)=2\int_{0}^{\infty}{\frac{e^{-at}sint}{t}}dt

\frac{dI}{da}=-2\int_{0}^{\infty}e^{-at}sint\: dt

\frac{dI}{da}=2\left[\frac{e^{-at}(asint+cost)}{a^{2}+1} \right]_{0}^{\infty}

\frac{dI}{da}=-\frac{2}{a^{2}+1}

I(a)=-2\, tan^{-1}a+c

as a tends to infinity integral tends to 0 : 0=-2\, tan^{-1}\infty + c

c=\pi

I(a)=-2\, tan^{-1}a+\pi

required integral : I(0)=\pi

Thanks Dubey Sir :)

uttara!!
he is much elder thn U.. see his profile.. so calling him dubey.. :-o

no problem