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Help me solve this integration WITHOUT using notion of Eo as it makes the problem to easy to solve and not understand it fully. Two

Help me solve this integration WITHOUT using notion of Eo as it makes the problem to easy to solve and not understand it fully.

Two infinite wires with the same positive lambda lineal charge density coincide with the x and y axes (cross shape). What is the resulting electric field at an arbitrary point in first quadrant (x, y)?

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