I am in a course currently studying this topic, and this is how I understand and solved the problem:
Electric field due to a continuous charge distribution:
E (as a vector) = \intkdq / r^2 (in r vector direction)
So, dE = kdq / r^2 = k\lambdadx / r^2
The E field has both x and y component, but if you draw the diagram as I see it, there is point p in the middle and directly above the uniformly charged line (ie along a line of symmetry). In that case, the x component will be 0 and there will just be the y.
Also, since this point is directly in the middle it cuts the charged line half, and so we let it's length (L) be L/2 until the intersection point and L/2 past the intersection point. This makes two right triangles with \Theta(1) representing the top angle for the first, and \Theta(2) representing the top angle for the second. It is difficult to explain this without a drawing...
So, we just need to find the y component of the E field:
E(y component) = dEcos\Theta = (k\lambdadx / r^2) (y / r) = k\lambdaydx/r^3
You can use trig substitution to solve this integral or use some relationships in the graph to simplify it. Notice cos\Theta(2) = y / r. So, 1/r = cos\Theta(2) / y.
Also, tan\Theta = x / y. So, x = ytan\Theta and dx = ysec^2\Theta d\Theta.
Plugging this stuff in we get dE(y) = k\lambdayysec^2\Thetad\Thetacos^3\Theta(2) / y^3
Simplifying should get k\lambdacos\Thetad\Theta / y
E(y) = k\lambda / y \intcos\Thetad\Theta
Solving this integral gives k\lambda / y (sin\Theta(2) - sin\Theta(1))
Since \Theta(2) = -\Theta(1) in the graph, the sines can be written as (sin\Theta - sin(-\Theta)) = 2sin\Theta
Thus, E(y) = (2k\lambda / y) sin\Theta
Notice from the graph that sin\Theta = x/r = (1/2L)/(\sqrt{(1/2L)^2 + y^2}
So, E(y) = (2k\lambda/y) ((1/2L)/(\sqrt{(1/2L)^2 + y^2})
The expression \lambdaL can be rewritten as Q, since it is the charge per unit length. So, on top, the 2 and (1/2) cancel leaving you with just kQ in the numerator.
Plugging in the numbers gives (8.99x10^9)(7x10^-6) / ((.69) * (\sqrt{1/2(.25)^2 + (.69)^2}))
The resulting calculation is 128042.6956 N/C
