Question
The equation $$\int_{ - \frac{\pi }{2}}^{\frac{\pi }{4}} {\left( {\lambda \left| {\sin \,x} \right| + \frac{{\mu \sin \,x}}{{1 + \cos \,x}} + \nu } \right)dx = 0,} $$ where $$\lambda ,\,\mu ,\,\nu $$ are constants, gives a relation between :
A.
$$\lambda ,\,\mu \,{\text{and }}\nu $$
B.
$$\lambda \,{\text{and }}\nu $$
C.
$$\lambda \,\,{\text{and }}\mu $$
D.
$$\mu \,\,{\text{and }}\nu $$
Answer :
$$\lambda \,{\text{and }}\nu $$
Solution :
$$\frac{{\mu \sin \,x}}{{1 + \cos \,x}}$$ is an odd function. So, $$\int_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{\mu \sin \,x}}{{1 + \cos \,x}}dx = 0.} $$
$$\therefore $$ the equation is $$\lambda \int_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\left| {\sin \,x} \right|dx + \nu \int_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {dx = 0} } $$
Or $$2\lambda \int_0^{\frac{\pi }{4}} {\left| {\sin \,x} \right|dx + \nu .\frac{\pi }{2} = 0,} $$ which is a relation between $$\lambda ,\,\nu .$$