E_x = 1/(4πε₀) ∫ λ dx * 1/ (Y² + x²) * x/((Y² + x²)^½
where Y is the vertical distance 15 cm.
λ dx is the charge element dq, 1/ (Y² + x²) is the " 1/r^2 " and x/((Y² + x²)^½ is the geometric factor for the x-component ("sin(α)" ).
The integral is easy and gives
E_x = λ/(8πε₀) ∫ du * 1/ u^(3/2)
= λ/(8πε₀) [-2/√u]
= λ/(4πε₀) (1/Y - 1/√(Y² + X²)) [where X = 32.0 cm]
Along the same line of reasoning we have for the y-component
E_y = 1/(4πε₀) ∫ λ dx * 1/ (Y² + x²) * Y/((Y² + x²)^½
= Yλ/(4πε₀) ∫ dx /(Y² + x²)^(3/2)
= λ/(4πε₀) X/(Y√(Y² + X²))
Just substitute λ (30*10^-9 C/m), X ( 0.30 m ) and Y ( 0.15 m) and ε₀ ( 8.854 10^-12 F/m) to calculate an uninteresting set of two numbers (E_x and E_y)