Calculate the double integral. \int \int_R \frac{4x}{1+xy}dA R=\left[0,4\right] \times \left[0,1\right]

coexpennan

coexpennan

Answered question

2021-05-12

Calculate the double integral.
R4x1+xydA
R=[0,4]×[0,1]

Answer & Explanation

toroztatG

toroztatG

Skilled2021-05-13Added 98 answers

Explanations for your question:

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Jeffrey Jordon

Jeffrey Jordon

Expert2021-09-30Added 2605 answers

Consider the double integral,

R4x1+xydA

Where R=[0,4]×[0,1]

Then R4x1+xydA=04014x1+xydydx

=044x[ln(1+xy)x]01dx

=044[ln|1+x|ln|1+0|]dx

=404[ln|1+x|0]dx

=404ln(1+x)dx

Now consider the integral 04ln(1+x)dx

Let u=ln(1+x),dv=dx

Then du=11+xdx, v=x

Use integration by parts,

udv=uvvdu

04ln(1+x)dx=[ln(1+x)xx11+xdx]04

=[xln(1+x)1+x11+xdx]04

=[xln(1+x)(xln|1+x|)]04

=[4ln53+ln5][00+0]

=5ln54 (2)

Use equation (2) in equation (1) to get,

R4x1+xydA=4[5ln54]

=20ln516

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