Question: Evaluate the following integral: ∫(x^3 + 2x^2 + 3x + 4) / (x^4 + 4x^3 + 6x^2 + 4x + 1) dx solve this integral

This integral is a bit tricky, but it can be solved by using the method of polynomial division and then simple integration.

Step 1: Polynomial Division We can rewrite the integral as ∫(x^3 + 2x^2 + 3x + 4) dx / (x^4 + 4x^3 + 6x^2 + 4x + 1). Now, we can perform polynomial division on the numerator and denominator. The result is 1 with a remainder of -x^3 - 2x^2 - 3x - 3.

Step 2: Rewrite the Integral Now, we can rewrite the integral as the sum of two separate integrals: ∫ dx - ∫(-x^3 - 2x^2 - 3x - 3) dx / (x^4 + 4x^3 + 6x^2 + 4x + 1).

Step 3: Solve the First Integral The first integral, ∫ dx, is simply x.

Step 4: Solve the Second Integral The second integral, ∫(-x^3 - 2x^2 - 3x - 3) dx / (x^4 + 4x^3 + 6x^2 + 4x + 1), is a bit more complicated. However, you might notice that the derivative of the denominator is -4x^3 - 12x^2 - 12x - 4, which is 4 times the numerator. Therefore, we can rewrite the integral as -1/4 ∫ du/u, where u = x^4 + 4x^3 + 6x^2 + 4x + 1. This integral is -1/4 ln|u|.

Step 5: Substitute Back for u Finally, we substitute back for u to get -1/4 ln|x^4 + 4x^3 + 6x^2 + 4x + 1|.

Step 6: Combine the Results The final result is x - 1/4 ln|x^4 + 4x^3 + 6x^2 + 4x + 1| + C, where C is the constant of integration.