The integral ∫dxx2+2x+5\int \frac{dx}{x^2+2x+5}∫x2+2x+5dx can be solved by completing the square as:
∫dx(x+1)2+4\int \frac{dx}{(x+1)^2 + 4}∫(x+1)2+4dx
∫dx(x+2)2+1\int \frac{dx}{(x+2)^2 + 1}∫(x+2)2+1dx
∫dx(x+1)2+2\int \frac{dx}{(x+1)^2 + 2}∫(x+1)2+2dx
∫dx(x+2)2+4\int \frac{dx}{(x+2)^2 + 4}∫(x+2)2+4dx