1. ## Re: Integration

Originally Posted by InteGrand
$\noindent Use polynomial division first and the remainder term'' (remainder you get from doing the polynomial long division divided by the divisor, x^2 + x +1 here) will be amenable to the procedure outlined above. This is because the remainder has lower degree than the divisor. (The quotient term will just be a polynomial, which is easy to integrate.)$

Note that in your previous one where I added and subtracted something to get the numerator of lower degree than the denominator was actually just a shortcut method of polynomial division. (So even if the numerator has equal degree to the denominator, we need to divide first so that the numerator degree becomes strictly less than the denominator degree, after which we can use the earlier outlined method.)

Also note that if the denominator is a quadratic (without square root) with two easy roots and the numerator is a linear function (including constant function), we can also use the method of partial fractions relatively easily.
What if the degree is the same but there is a square root on the bottom?

2. ## Re: Integration

Prolly a trig substitution if x^2 sorta thing. A subsitution will also work well

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