Question

The equation ${\int }_{-\pi /4}^{\pi /4}(\lambda \left|\sin x\right|+\frac{\mu \sin x}{1+\cos x}+v)dx=0$,where$\lambda ,\mu ,v$ are constants,

gives a relation between ?

• A. $\lambda ,\mu$ and $v$
• B. $\lambda$ and $v$
• C. $\lambda ,\mu$
• D. $\mu$ and $v$

Answer 1

Answer: (B)

$\lambda$ and $v$

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}(\lambda \left|\sin x\right|+\frac{\mu \sin x}{1+\cos x}+v)dx$
$={\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\left|\sin x\right|+\mu {\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{\sin x}{1+\cos x}dx+v{\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}dx$
$=-\lambda {\int }_{-\frac{\pi }{4}}^0\sin xdx+\lambda {\int }_0^{\frac{\pi }{4}}\sin x+\mu {\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{\sin x}{1+\cos x}dx+v{\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}dx$
$=-\lambda {[-\cos x]}_{-\frac{\pi }{4}}^0+\lambda {[-\cos x]}_0^{\frac{\pi }{4}}+\mu {[-\log (1+\cos x)]}_{-\frac{\pi }{4}}^{\frac{\pi }{4}}+v{\left[x\right]}_{-\frac{\pi }{4}}^{\frac{\pi }{4}}$
$=-\lambda (-1+\frac{1}{\sqrt{2}})+\lambda (\frac{-1}{\sqrt{2}}+1)$
$+\mu [-\log (1+\frac{1}{\sqrt{2}})+\log (1+\frac{1}{\sqrt{2}})]+v[\frac{\pi }{4}+\frac{\pi }{4}]$
$=2\lambda +v\frac{\pi }{2}$