Question Number 213656 by efronzo1 last updated on 12/Nov/24
Answered by golsendro last updated on 13/Nov/24
![(i) g(4−x)= −g(x) ∫_0 ^4 g(x)dx = ∫_0 ^4 g(4−x)dx ∫_0 ^4 −g(4−x)dx = ∫_0 ^4 g(4−x)dx it folow that ∫_0 ^4 g(4−x)dx= 0 and ∫_0 ^4 g(x)dx = 0 (ii) f(x^2 )= f((−x)^2 ) , f(x) even function ∫_(−4) ^4 f(x^2 )dx = 2∫_0 ^4 f(x^2 )dx = 2 ∫_0 ^4 (4x^3 −g(x))dx = 2∫_0 ^4 4x^3 dx−0 = 2[ x^4 ]_0 ^4 = 2×256= 512](https://www.tinkutara.com/question/Q213657.png)
$$\:\:\left(\mathrm{i}\right)\:\mathrm{g}\left(\mathrm{4}−\mathrm{x}\right)=\:−\mathrm{g}\left(\mathrm{x}\right)\: \\ $$$$\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{g}\left(\mathrm{4}−\mathrm{x}\right)\mathrm{dx}\: \\ $$$$\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}−\mathrm{g}\left(\mathrm{4}−\mathrm{x}\right)\mathrm{dx}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\mathrm{g}\left(\mathrm{4}−\mathrm{x}\right)\mathrm{dx} \\ $$$$\:\:\:\:\:\:\mathrm{it}\:\mathrm{folow}\:\mathrm{that}\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{g}\left(\mathrm{4}−\mathrm{x}\right)\mathrm{dx}=\:\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{and}\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\mathrm{0} \\ $$$$\:\left(\mathrm{ii}\right)\:\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)=\:\mathrm{f}\left(\left(−\mathrm{x}\right)^{\mathrm{2}} \right)\:,\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{even}\:\mathrm{function} \\ $$$$\:\:\:\underset{−\mathrm{4}} {\overset{\mathrm{4}} {\int}}\:\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx}\:=\:\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\:\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\left(\mathrm{4x}^{\mathrm{3}} −\mathrm{g}\left(\mathrm{x}\right)\right)\mathrm{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:\mathrm{4x}^{\mathrm{3}} \:\mathrm{dx}−\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\left[\:\mathrm{x}^{\mathrm{4}} \:\right]_{\mathrm{0}} ^{\mathrm{4}} \:=\:\mathrm{2}×\mathrm{256}=\:\mathrm{512} \\ $$