Integral from Zero to Zero?

Calculus Level 4

$\lim_{k\to{0}} \int_{0}^{k} \frac{1}{\sqrt{\cos(x)-\cos(k)}} dx = \ ?$

\frac{\pi^2}{3}

0

\sqrt{\frac{{\pi}}{2}}

1

\frac{\pi}{\sqrt{2}}

\frac{\pi}{4}

2 solutions

MagmaUnleashed .
Feb 14, 2020

I= \lim {k \rightarrow 0} \int 0^k 1 / \sqrt[2]{cosk-cosx} dx

                              let x=kt \Longrightarrow dx=kdt

I= \lim {k \rightarrow 0} \int 0^k k/ \sqrt{coskt-cosx}dt

              from the taylor expansion of cosx we get,

cosx=1-x^2/2!+x^4/4!..... \Longrightarrow cosx=1-x^2/2 at x=0

I= \int 0^1 \sqrt{2}k/ \sqrt{k^2(1-t^2)}dt= \int 0^1 \sqrt{2}/ \sqrt{1-t^2}dt

I= \sqrt{2}arcsint= \sqrt{2} \Pi /2= \Pi / \sqrt{2}

It is not clear

Shrujeeth Kshatriya - 1 year, 2 months ago

please use latex

Ajay Anand Dwivedi - 10 months, 3 weeks ago

Neel Shelgaonkar
Feb 20, 2020

If u choose such a k that it is infinitesimally small the difference of cosine is product of sine and that product can be approximated as sin({x-k}/2) as x-k/2 and similar for the other

Integral from Zero to Zero?

2 solutions

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