Question

need help with this definite integral : (posting the question below)

Answer 1

\[\frac{ dy }{ dx } = \sin (x+y) + \cos (x+y) -1\]

the answer given is

\[\log_{} [cosec (x+y+ \frac{\pi}{4}) - \cot (x+y+ \frac{\pi}{4}) ] = \sqrt{2}x +c\]

what i have done : i took x+y =z
the equation then became

\[\frac{ dz }{ dx } = \sin z - \cos z\]

or
\[\int\limits\limits \frac {dz}{\sin z + \cos z} = \int\limits\limits dx\]

can't proceed any more..please help

Answer 2

Hint :
\[\sin z + \cos z = \sqrt{2} \left(\dfrac{1}{\sqrt{2}}\sin z + \dfrac{1}{\sqrt{2}}\cos z\right) = \sqrt{2}\sin\left(z + \frac{\pi}{4}\right)\]

Answer 3

so it becomes
\[\ \frac{1} {\sqrt{2}}\ \int\limits \csc\left(z + \frac{\pi}{4}\right) dz ?? \]

Answer 4

Yep!