#intint_D cosxy dxdy#, where #D: {[x>=0],[y>=0],[x>=y]}#?
1 Answer
This integration is in the first quadrant, under the line
Following the blue integration:
For that last integration, a sketch:
Create function:
#I(a ) = int_0^oo dz qquad (sin z)/z e^(- a z) qquad triangle qquad a>=0#
Differentiate wrt
#I'(a ) = int_0^oo dz qquad - z (sin z)/z \ e^(- a z)#
That, after 2 rounds of IBP, is:
Integrate wrt
Combining:
-
#triangle: qquad lim_(a to oo) int_0^oo dz qquad (sin z)/z e^(- a z) = 0# -
#square: qquad I(a to oo) = - arctan (a to oo) + C = 0#
And:
#I(0) = int_0^oo dz qquad (sin z)/z = pi/2#