3.5.4—An integrand of the form x

For ∫x/(ax²+bx+c)dx, write the numerator as a multiple of the denominator derivative plus a constant remainder. The first part gives a logarithm; the remainder may require completing the square.

Differentiate the denominator first, solve a simple linear identity for the numerator split, and keep the denominator positive conditions for logs.

For ∫x/(x²+1)dx, x is half of (x²+1)′, so the result is ½ln(x²+1)+C.

The numerator need not equal the denominator derivative exactly; it may be a scaled derivative plus a leftover constant.