definite integral with limit

[stuart clark]

\(\displaystyle \displaystyle\lim_{n \to \infty}n.\int_{0}^{\frac{\pi}{2}}\left(1-\sqrt[n]{sinx}\right)dx\)

 

[galactus]

Note that the limit of the integral is the integral of the limit.

So, taking the limit of \(\displaystyle n(1-\sqrt[n]{sin(x)})\).

By L'Hopital , we get \(\displaystyle \lim_{n\to \infty}-sin^{\frac{1}{n}}(x)ln(sin(x))\)

\(\displaystyle sin^{\frac{1}{n}}(x)\rightarrow 1\) as \(\displaystyle n\to \infty\).

So, we end up with \(\displaystyle -ln(sin(x))\)

Now, integrate:

\(\displaystyle -\int_{0}^{\frac{\pi}{2}}ln(sin(x))dx\)

This is a rather famous, but tough, integral. You can find the solution by Googling.

If not, let me know and maybe we can step through it.

Start by letting \(\displaystyle sin(x)=2sin(x/2)cos(x/2)\).

 

[stuart clark]

thanks galactus.