Compute the definite integral: ∫0−2log⁡(|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=fg−∫gdf, where f=log⁡(−x),dg=dx,df=1xdx,g=x: =(−xlog⁡(−x))|−20+∫−201dx Evaluate the antiderivative at the limits and subtract. limb→0−(−xlog⁡(−x))|−2b=(limb→0−−log⁡(−b)b)−(−(−2)log⁡(−(−2)))=(limb→0−−log⁡(−b)b)−log⁡(4): =(limb→0−−log⁡(−b)b)−log⁡(4)+∫−201dx limb→0−b(−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: =2−log⁡(4)

Solved: "∫0−2ln|x|dx"

Marvin Mccormick

Answered question

2021-09-12

$\int_{0}^{- 2} \ln | x | d x$

Answer & Explanation

Nathanael Webber

Skilled2021-09-13Added 117 answers

Compute the definite integral:
$\int_{0}^{- 2} \log (| x |) d x$
Switch the order of the integration bounds of $\log (| x |)$ so that the upper bound is larger. Multiply the integrand by -1:
$= - \int_{- 2}^{0} \log (| x |) d x$
Simplify $\log (| x |)$ assuming -2 $= - \int_{- 2}^{0} \log (- x) d x$
For the integrand log(-x), integrate by parts, integral $f d g = f g - \int g d f$ , where
$f = \log (- x), d g = d x, d f = \frac{1}{x} d x, g = x$ :
$= (- x \log (- x)) |_{- 2}^{0} + \int_{- 2}^{0} 1 d x$
Evaluate the antiderivative at the limits and subtract.
$lim_{b \to 0^{-}} (- x \log (- x)) |_{- 2}^{b} = (lim_{b \to 0^{-}} - \log (- b) b) - (- (- 2) \log (- (- 2))) = (lim_{b \to 0^{-}} - \log (- b) b) - \log (4)$ :
$= (lim_{b \to 0^{-}} - \log (- b) b) - \log (4) + \int_{- 2}^{0} 1 d x$
$lim_{b \to 0^{-}} b (- \log (- b)) = 0$ :
$= - \log (4) + \int_{- 2}^{0} 1 d x$
Apply the fundamental theorem of calculus.
The antiderivative of 1 is x:
$= - \log (4) + x |_{- 2}^{0}$
Evaluate the antiderivative at the limits and subtract.
$x |_{- 2}^{0} = 0 - (- 2) = 2$ :
Answer:
$= 2 - \log (4)$

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