Question

integrate

Answer 1

\[put a=r \cos \alpha,b=r \sin \alpha \]
square and add
\[a ^{2}+b ^{2}=r ^{2}\left( \cos ^{2}\alpha+\sin ^{2}\alpha \right)=r ^{2}\]
\[r=\sqrt{a ^{2}+b ^{2}},\tan \alpha=\frac{ b }{ a },\alpha=\tan^{-1} \frac{ b }{ a }\]
\[I=\int\limits \frac{ dx }{r \sin \left( x+\alpha \right) } =\frac{ 1 }{ r }\int\limits \frac{ dx }{2\sin \frac{ x+\alpha }{2 }\cos \frac{ x+\alpha }{2 } }\]
divide the numerator and denominator by\[\cos ^{2}\frac{ x+\alpha }{ 2 } and get the solution.\]

Answer 2

i didnt get the last step why we need to divide
cos^2(x+alpha/2)

Answer 3

\[I=\frac{ 1 }{ r }\int\limits \frac{ \sec ^{2}\frac{ x+\alpha }{ 2 }\frac{ 1 }{2}dx }{\tan \frac{ x+\alpha }{2} }\]
\[=\frac{ 1 }{r }\ln \tan \frac{ x+\alpha }{ 2 }+c\]
Replace the values of r and alpha.