Menu Close

1-1-1-y-2-1-y-2-ln-x-2-y-2-1-dx-dy-




Question Number 190419 by horsebrand11 last updated on 02/Apr/23
 ∫_(−1) ^1 ∫_(−(√(1−y^2 ))) ^(√(1−y^2 ))  ln (x^2 +y^2 +1)dx dy =?
111y21y2ln(x2+y2+1)dxdy=?
Answered by witcher3 last updated on 04/Apr/23
x=rcos(a)  y=rsin(a)  −1≤rsin(a)≤1,⇒a∈[−(π/2),(π/2)]  ∣x∣≤(√(1−y^2 ))   ⇔0≤x^2 +y^2 ≤1⇔0≤r≤1  −(1/r)≤sin(a)≤(1/r)  ∫_0 ^1 ∫_(−(π/2)) ^(π/2) rln(1+r^2 )dadr=(π/2)∫_0 ^1 ln(1+r^2 )dr^2   =(π/2)∫_0 ^1 ln(1+x)dx=(π/2).(2ln(2)−1)
x=rcos(a)y=rsin(a)1rsin(a)1,a[π2,π2]x∣⩽1y20x2+y210r11rsin(a)1r01π2π2rln(1+r2)dadr=π201ln(1+r2)dr2=π201ln(1+x)dx=π2.(2ln(2)1)

Leave a Reply

Your email address will not be published. Required fields are marked *