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Question Number 132287 by benjo_mathlover last updated on 13/Feb/21

Answered by Olaf last updated on 13/Feb/21

$$\mathrm{Let}q=2{\int }_0^{6}f\left(x\right)dx=2{\int }_0^{6}f(x-4)dx\left(1\right)$$$$\mathrm{Let}u=x-4$$$$\left(1\right):q=2{\int }_{-4}^{2}f\left(u\right)du$$$$q=2{\int }_{-4}^{-2}f\left(u\right)du+2{\int }_{-2}^{0}f\left(u\right)du+2{\int }_0^{2}f\left(u\right)du$$$$f\mathrm{is}\mathrm{odd}:$$$$q=-2{\int }_2^{4}f\left(u\right)du-2{\int }_0^{2}f\left(u\right)du+2{\int }_0^{2}f\left(u\right)du$$$$q=-2{\int }_2^{4}f(u-4)du$$$$\mathrm{Let}v=u-4$$$$q=-2{\int }_{-2}^{0}f\left(v\right)dv$$$$f\mathrm{is}\mathrm{odd}:$$$$q=2{\int }_0^{2}f\left(v\right)dv$$$$q=2p$$

Commented by Olaf last updated on 13/Feb/21

$$sorrymister,Icorrected$$

Commented by benjo_mathlover last updated on 13/Feb/21

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Answered by Ñï= last updated on 14/Feb/21

$$2{\int }_0^{6}f\left(x\right)dx$$$$=2{\int }_0^{6}f(x-4)dx$$$$=2{\int }_{-4}^{2}f\left(x\right)dx$$$$=2({\int }_{-2}^{2}f\left(x\right)dx+{\int }_{-4}^{-2}f\left(x\right)dx)$$$$=2{\int }_0^{2}f(x-4)dx$$$$=2{\int }_0^{2}f\left(x\right)dx$$$$=2p$$

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