Question

What is `int (-x^3-2x-3 ) / (7x^4+ 5 x -1 )`?

Answer 1

`-.65697ln|x-.19785|-.01191ln|x+.95254|-.51407ln|.78462/sqrt(x^2-.75468x+.75801)|-.05036tan^(-1)((x-.37734)/.78462)+const.`

Explanation:

What are the roots of the polynomial
`7x^4+5x-1`?

This is a depressed quartic function and I recommend this 2 sources (the first is to show work, the second for a fast solution) to resolve it:

http://www.sosmath.com/algebra/factor/fac12/fac12.html

http://www.mathportal.org/calculators/polynomials-solvers/polynomial-roots-calculator.php

From the root finder we get:

`x_1=.19785`
`x_2=-.95254`
`x_3=.37734+i.78462`
`x_4=.37734-i.78462`

For not to deal with complex numbers we have
`(x-x_3)(x-x_4)=x^2-.75468x+.75801`

The original function can be rewritten in partial fractions in this way
`(-x^3-2x-3)/(7x^4+5x-1)=A/(x-.19785)+B/(x+.95254)+(Cx+D)/(x^2-.75468x+.75801)`

For `x=0, -1, 1 and 2` we get

`A/-.19785+B/.95254+0*C+D/.75468=3`
`A/-1.19785+B/.04746+(-C+D)/5.51269=0`
`A/.80215+B/1.95254+(C+D)/1.00333=-6/11`
`A/1.80295+B/2.95254+(2C+D)/3.24865=-15/121`

Or

`[[-5.05433,1.04982,0,1.31924],[-.83483,21.07038,-.39798,.39798],[1.24665,.51215,.99668,.99668],[.55489,.33869,.61564,.30782]][[A],[B],[C],[D]]=[[3],[0],[-6/11],[-15/121]]`

Solving this system of variables we get

`A=-.65697`
`B=-.01191`
`C=.51407`
`D=-.23349`

So the original expression becomes

`-.65697int dx/(x-.19785)-.01191int dx/(x+.95254)+int (.51407x-.23349)/(x^2-.75468x+.75801)dx` [ `alpha` ]

Let's resolve the last part of the expression, the only one that poses a challenge
`int (.51407x-.23349)/(x^2-.75468x+.75801)dx=`

`(x-.37734)^2=x^2-.75468+.14239`
=> `x^2-.75468x+.75801=(x-.37734)^2+.61562`
`(x-.37734)=sqrt(.61562)tany=.78462tany`
`dx=.78462sec^2ydy`
How many units of `(x-.37734)` are there in the numerator?
`(.51407x-.23349)/(x-.37734)=.51407-.03951/(x-.37734)`

That's why now we are dealing with
`=.51407int (.78462tany*.78462cancel(sec^2y))/(.61562cancel(sec^2y))dy-.03951int (.78462cancel(sec^2y))/(.61562cancel(sec^2y)dy`
`=.51407int tanydy-.05036int dy`
`=-.51407ln |cosy|-.05036y`

`tany=(x-.37734)/.78462` => `siny=(x-.37734)/.78462cosy`
=>`sin^2y+cos^2y=1` => `((x^2-.75468x+.14239)/.61562+1)cos^2 y=1` => `cosy=.78462/sqrt(x^2-.75468x+.75801)`
`-> =-.51407 ln |.78462/sqrt(x^2-.75468x+.75801)|-.05036tan^(-1)((x-.37734)/.78462)`

Therefore expression [ `alpha` ] becomes
`-.65697ln|x-.19785|-.01191ln|x+.95254|-.51407ln|.78462/sqrt(x^2-.75468x+.75801)|-.05036tan^(-1)((x-.37734)/.78462)+const.`