This "The Question ∫csc⁡(x−π3)csc⁡(x−π6)dx What I Tried- I tried..." has been explained

Secondary
Geometry
Trigonometry
Trigonometric equations and identities

Lorraine Harvey Answered2021-12-31

The Question

\int \csc (x - \frac{π}{3}) \csc (x - \frac{π}{6}) d x

What I Tried- I tried dividing both numerator and denominator by

\sin \frac{π}{6}

but couldnt

Answer & Explanation

boronganfh

Expert

2022-01-01Added 33 answers

Hint

\csc (x - \frac{π}{3}) \csc (x - \frac{π}{6})

= \frac{1}{\sin (\frac{π}{3} - \frac{π}{6})} \cdot \frac{\sin (x - \frac{π}{6} - (x - \frac{π}{3}))}{\sin (x - \frac{π}{3}) \sin (x - \frac{π}{6})}

hysgubwyri3

Expert

2022-01-02Added 43 answers

\csc (x - \frac{π}{3}) \csc (x - \frac{π}{6}) = \csc (\frac{π}{6} - x) \sec (x + \frac{π}{6}) = \frac{4}{\sqrt{3} - 2 \sin (2 x)}

Now, let

x = \tan^{- 1} (t)

and tou face

I = 4 \int \frac{d t}{\sqrt{3} t^{2} - 4 t + \sqrt{3}}

The denominator has two simple real roots. Then partial fraction decomposition to face two simple integrals.

Vasquez

Expert

2022-01-08Added 457 answers

Substitute $t = \tan (x - \frac{π}{4})$
$\int \csc (x - \frac{π}{3}) \csc (x - \frac{π}{6}) d x = 4 \int \frac{2 - \sqrt{3}}{t^{2} - (2 - \sqrt{3})^{2}} d t$
$= 4 \ln | \frac{2 - \sqrt{3} - t}{2 - \sqrt{3} + t} | = 4 \ln | \frac{2 - \sqrt{3} - \tan (x - \frac{π}{4})}{2 - \sqrt{3} + \tan (x - \frac{π}{4})} | + C$