- #1
joemama69
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Homework Statement
show what integral of 1/(1-xy)dxdy = pi2/6
dx 0 to 1
dy 0 to 1
joemama69 said:[tex]\int[/tex]1/(1-xy)dx = ln(1-xy) = ln(1-y) - ln1
[tex]\int[/tex]ln(1-y) - ln1 dy = (1-y)ln(1-y) - (1- y) - yln1
= -ln1 - (ln1 - 1) = = -2ln1 + 1 huh
The integral of 1/(1-xy) is Pi²/6 from 0 to 1.
The integral of 1/(1-xy) can be solved using the substitution method, where u = 1-xy and du = -xdy. This results in the integral becoming -∫(1/u)du, which can then be solved using basic integration rules.
The limits 0 to 1 represent the range of values for which the integral is being evaluated. In this case, the integral is being evaluated from 0 to 1, which means the area under the curve is being calculated for the interval between 0 and 1 on the x-axis.
The result of the integral of 1/(1-xy) in terms of Pi is due to the use of trigonometric identities in the integration process. These identities involve Pi and result in the final answer being expressed in terms of Pi.
Yes, the integral of 1/(1-xy) can also be solved using partial fractions or by using the method of integration by parts. However, the substitution method is the most commonly used and efficient method for solving this integral.