manipulation of numerator in terms of the denominator

Integration by manipulation of numerator in terms of the denominator

Integration formulae

(px+q) / (ax² + bx +c) , (px+q) /sqrt(ax² + bx +c) ,

(px² + qx +r) /(ax² + bx +c) type

(psinx+qcosx ) / (a sinx + b cosx )

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for

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you might then have to restore to completion of squares in some cases after the manipulation
in some cases the manipulation can be done through inspection

Examples

* ∫ { (4x - 7) / (x² + x +1)} dx
integration of linear expression in numerator mixed with quadratic expression

* ∫ { cosx/ [ sinx +cosx] } dx
integral of [ cos x ] / [sin x + cos x ]

*not the above types but still an example of manupulating the angle in the numerator in terms of the angles in the denominator


integral of 1 / [cos(x+a)cos(x+b)]

*integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ]  }
answer and a little explanation of integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ]  }



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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