integral of linear expression in the numerator mixed with a quadratic expression

integral of  ∫ { (4x - 7) / (x² + x +1)} dx

here there is a linear expression in the numerator mixed with quadratic expression containing both (x^2)  and x terms

try to express the linear expression in terms of the derivative of the quadratic expression and a constant.
then break the numerator into two parts.
 one will be of the form [f '(x) / f(x)] , for the second part use completion of squares.

 

PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION 

PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

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