{\int_{{0}}^{{-{2}}}}{\ln}{\left|{x}\right|}{\left.{d}{x}\right.}

Marvin Mccormick

Marvin Mccormick

Answered question

2021-09-12

02ln|x|dx

Answer & Explanation

Nathanael Webber

Nathanael Webber

Skilled2021-09-13Added 117 answers

Compute the definite integral:
02log(|x|)dx
Switch the order of the integration bounds of log(|x|) so that the upper bound is larger. Multiply the integrand by -1:
=20log(|x|)dx
Simplify log(|x|) assuming -2 =20log(x)dx
For the integrand log(-x), integrate by parts, integral fdg=fggdf, where
f=log(x),dg=dx,df=1xdx,g=x:
=(xlog(x))|20+201dx
Evaluate the antiderivative at the limits and subtract.
limb0(xlog(x))|2b=(limb0log(b)b)((2)log((2)))=(limb0log(b)b)log(4):
=(limb0log(b)b)log(4)+201dx
limb0b(log(b))=0:
=log(4)+201dx
Apply the fundamental theorem of calculus.
The antiderivative of 1 is x:
=log(4)+x|20
Evaluate the antiderivative at the limits and subtract.
x|20=0(2)=2:
Answer:
=2log(4)

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