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Which of the following integrals represents the area of the region bounded by x = e and the functions f(x) = ln(x) and g(x) = log1/e(x)?

the integral from 0 to e of the quantity, natural log of x minus the log base 1 over e of x, dx

the integral from 1 to e of the quantity, the log base 1 over e of x minus the natural log of x, dx

the integral from negative 1 to 0 of 4 minus the log base 1 over e of x,dx plus the integral from 0 to 1 of 4 minus the natural log of x, dx

the integral from 1 to e of the quantity, natural log of x minus the log base 1 over e of x, dx

Respuesta :

LammettHash

Notice that

[tex]\log_{1/e}x=\dfrac{\ln x}{\ln\frac1e}=\dfrac{\ln x}{-\ln e}=-\ln x[/tex]

[tex]f(x)=\ln x[/tex] and [tex]g(x)=-\ln x[/tex] intersect when [tex]x=1[/tex]. For all [tex]x>1[/tex], we have [tex]\ln x>0[/tex] and [tex]-\ln x<0[/tex], so [tex]f(x)>g(x)[/tex]. Then the area we want is given by the integral,

[tex]\displaystyle\int_1^e\ln x-(-\ln x)\,\mathrm dx=2\int_1^e\ln x\,\mathrm dx[/tex]

or in terms of [tex]\log_{1/e}x[/tex],

[tex]\displaystyle\int_1^e\ln x-\log_{1/e}x\,\mathrm dx[/tex]

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