3
$\begingroup$

The integral was: $$\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\frac{\pi + 4x^6}{1-\sin(|x|+\frac{\pi}{6})}$$

What I did was to identify that its an even function and write it as: $$2\int_{0}^{\frac{\pi}{6}}\frac{\pi + 4x^6}{1-\sin(x+\frac{\pi}{6})}$$

Then I wrote the denominator in terms of $\cos$ as $$\sin(x+\frac{\pi}{6})=\cos(\frac{\pi}{3}-x)$$ and then I applied the identity $1-\cos x=2\sin^2(\frac{x}{2})$ to finally get :

$$2\biggl[\biggl(\int_{0}^{\frac{\pi}{6}}\frac{\pi}{2\sin^2(\frac{\pi}{6}-\frac{x}{2})}dx\biggl)+\biggl(\int_{0}^{\frac{\pi}{6}}\frac{4x^6}{2\sin^2(\frac{\pi}{6}-\frac{x}{2})}dx\biggl)\biggl]$$

Now the first half is easy to get but how do I integrate the second one? Can someone suggest any steps or perhaps an alternative way?

$\endgroup$
1
  • 1
    $\begingroup$ I guess there's not a beautiful closed form for your integral. I tried to use mathematica to compute and it gives the answer $$\frac{10}{27} i \pi ^4 \text{Li}_2\left(\sqrt[6]{-1}\right)+\frac{80}{9} \pi ^3 \text{Li}_3\left(\sqrt[6]{-1}\right)-160 i \pi ^2 \text{Li}_4\left(\sqrt[6]{-1}\right)+\pi \left(4-1920 \text{Li}_5\left(\sqrt[6]{-1}\right)\right)+11520 i \left(\text{Li}_6\left(\sqrt[6]{-1}\right)-\text{Li}_6\left(\frac{1}{2} \left(i \sqrt{3}+1\right)\right)\right)+\frac{\pi ^6 \left(\sqrt{3}+(2+i)\right)}{5832}+\frac{1}{81} \pi ^5 \log \left(1-\sqrt[6]{-1}\right)$$ $\endgroup$
    – FFjet
    Sep 11, 2019 at 5:25

1 Answer 1

1
$\begingroup$

Looking at the monster given by Mathematica in comments, consider $$I=\int_{0}^{\frac{\pi}{6}}\frac{\pi + 4x^6}{1-\sin(x+\frac{\pi}{6})}\,dx=\pi\int_{0}^{\frac{\pi}{6}}\frac{dx}{1-\sin(x+\frac{\pi}{6})}+4\int_{0}^{\frac{\pi}{6}}\frac{x^6}{1-\sin(x+\frac{\pi}{6})}\,dx$$ The first one is simple.

For the second one, why not to try series expansion around $x=0$ $$\frac{1}{1-\sin(x+\frac{\pi}{6})}=2+2 \sqrt{3} x+5 x^2+\frac{11 x^3}{\sqrt{3}}+\frac{91 x^4}{12}+\frac{301 x^5}{20 \sqrt{3}}+O\left(x^6\right)$$ which, integrated termwise would give $$\int_{0}^{\frac{\pi}{6}}\frac{x^6}{1-\sin(x+\frac{\pi}{6})}\,dx\approx 0.00902$$ while the numerical integration would give $\approx 0.00937$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.