Question and Answers Forum

All Questions      Topic List

UNKNOWN Questions

Previous in All Question      Next in All Question      

Previous in UNKNOWN      Next in UNKNOWN      

Question Number 52949 by gunawan last updated on 15/Jan/19

$$\mathrm{If}f\left(x\right)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{function},\mathrm{then}$$$$\underset{0}{\stackrel{\pi }{\int }}f\left(\cos x\right)dx=2\underset{0}{\stackrel{\pi /2}{\int }}f\left(\cos x\right)dx$$

Commented by tanmay.chaudhury50@gmail.com last updated on 15/Jan/19

$$recheckthequestion...$$

Commented by maxmathsup by imad last updated on 15/Jan/19

$${\int }_0^{\pi }f\left(cosx\right)dx={\int }_0^{\frac{\pi }{2}}f\left(cosx\right)dx+{\int }_{\frac{\pi }{2}}^{\pi }f\left(cosx\right)dxbut$$$${\int }_{\frac{\pi }{2}}^{\pi }f\left(cosx\right)dx{=}_{x=\frac{\pi }{2}+t}{\int }_0^{\frac{\pi }{2}}f(-sint)dt=-{\int }_0^{\frac{\pi }{2}}f\left(sint\right)dt\left(fisodd\right)$$$$=-{\int }_0^{\frac{\pi }{2}}f\left(cos(\frac{\pi }{2}-t)\right)dt{=}_{\frac{\pi }{2}-t=u}-{\int }_{\frac{\pi }{2}}^{0}f\left(cosu\right)(-du)$$$$=-{\int }_0^{\frac{\pi }{2}}f\left(cosu\right)du\Rightarrow {\int }_0^{\pi }f\left(cosx\right)dx=0$$$$iffisevenweget{\int }_{\frac{\pi }{2}}^{\pi }f\left(cost\right)dt{=}_{x=\frac{\pi }{2}+t}{\int }_0^{\frac{\pi }{2}}f(-sint)dt={\int }_0^{\frac{\pi }{2}}f\left(sint\right)dt$$$$={\int }_0^{\frac{\pi }{2}}f\left(cos(\frac{\pi }{2}-t)\right)dt{=}_{\frac{\pi }{2}-t=u}-{\int }_{\frac{\pi }{2}}^{0}f\left(cosu\right)du={\int }_0^{\frac{\pi }{2}}f\left(cosu\right)du\Rightarrow$$$${\int }_0^{\pi }f\left(cosx\right)dx=2{\int }_0^{\frac{\pi }{2}}f\left(cosx\right)dxsothereisaerrorinthequestion!...$$

Commented by gunawan last updated on 16/Jan/19

$$\mathrm{thank}\mathrm{you}\mathrm{Sir}$$

Commented by maxmathsup by imad last updated on 16/Jan/19

$$youarewelcome$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com