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Question Number 209544 by Spillover last updated on 13/Jul/24

                      ∫_0 ^4 (dx/(∣x−1∣+∣x−2∣+∣x−4∣))

$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{4}} \frac{{dx}}{\mid{x}−\mathrm{1}\mid+\mid{x}−\mathrm{2}\mid+\mid{x}−\mathrm{4}\mid} \\ $$$$ \\ $$

Answered by Frix last updated on 14/Jul/24

=∫_0 ^1 (dx/(7−3x))+∫_1 ^2 (dx/(5−x))+∫_2 ^4 (dx/(x+1))=  =((ln 7)/3)+ln 5 −2ln 3 +(4/3)ln 2

$$=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{dx}}{\mathrm{7}−\mathrm{3}{x}}+\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}\frac{{dx}}{\mathrm{5}−{x}}+\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\frac{{dx}}{{x}+\mathrm{1}}= \\ $$$$=\frac{\mathrm{ln}\:\mathrm{7}}{\mathrm{3}}+\mathrm{ln}\:\mathrm{5}\:−\mathrm{2ln}\:\mathrm{3}\:+\frac{\mathrm{4}}{\mathrm{3}}\mathrm{ln}\:\mathrm{2} \\ $$

Commented by Spillover last updated on 30/Jul/24

great

$${great} \\ $$

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