Integration by Partial Fractions

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To integrate the expression (x^3 + 49) / (x^2 + 5x + 4), the first step is to perform polynomial long division since the degree of the numerator is higher than that of the denominator. After dividing, the result will include a polynomial term and a remainder that can be expressed over the factored denominator, which is (x + 4)(x + 1). Simplifying the remainder will allow for the application of partial fraction decomposition. This process will lead to a more manageable integral that can be solved using standard integration techniques. The key steps involve division, factoring, and then applying partial fractions to complete the integration.
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Homework Statement


Integrate

x^3 + 49 / x^2 + 5x + 4


Homework Equations





The Attempt at a Solution


Since the numerator has an x cubed, but the denominator only has an x squared, I know I need to divide the numerator by something.

I'm not sure what, but maybe the denominator?

At some point I know I need to factor the denominator, and I got (x+4)(x+1).

Once I get that top part simplified I think I know what to do, I'm just not sure how to simplify.
 
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Yes, divide the numerator by the denominator. You'll get x + (other terms)/(x^2 + 5x + 4).
 

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