The Question ∫csc⁡(x−π3)csc⁡(x−π6)dx What I Tried- I tried dividing both numerator and denominator by sin⁡π6...

Lorraine Harvey

Lorraine Harvey

Answered

2021-12-31

The Question
csc(xπ3)csc(xπ6)dx
What I Tried- I tried dividing both numerator and denominator by sinπ6 but couldnt

Answer & Explanation

boronganfh

boronganfh

Expert

2022-01-01Added 33 answers

Hint
csc(xπ3)csc(xπ6)
=1sin(π3π6)sin(xπ6(xπ3))sin(xπ3)sin(xπ6)
hysgubwyri3

hysgubwyri3

Expert

2022-01-02Added 43 answers

csc(xπ3)csc(xπ6)=csc(π6x)sec(x+π6)=432sin(2x)
Now, let x=tan1(t) and tou face
I=4dt3t24t+3
The denominator has two simple real roots. Then partial fraction decomposition to face two simple integrals.
Vasquez

Vasquez

Expert

2022-01-08Added 457 answers

Substitute t=tan(xπ4)
csc(xπ3)csc(xπ6)dx=423t2(23)2dt
=4ln|23t23+t|=4ln|23tan(xπ4)23+tan(xπ4)|+C

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