Frequently Requested Easy Hard Integrals


...In response to a student's e-mail...

>
> Professor Ritter,
>     I am having trouble figuring out the integral of (x-1)/(x+2)*dx.
> Also the integral of x/(x+1)*dx.  I checked my Cal book and couldn't
> find it.  Could you help me out, it is killing me......  Also just to
> make sure, Tuesday's test is only up to section Y correct?  I missed
> the last lecture so I want to make sure nothing changed.  Thank you.
>
>
> ---------------------------------

Mr. X,

Who taught you Calculus II??  The test on Tuesday is _through_ Y NOT
_up_ _to_ Y.  Both integrands of

		S (x-1)/(x+2) dx and  S x/(x+1) dx

are RATIONAL FUNCTIONS that are not PROPER FRACTIONS because the degree of
the numerator is >= the degree of the denominator.  Do long division
first.  The rational function may then be written in the form

                               Remainder(x)
                Quotient(x) +  ----------- .
                               Divisor(x)

So for the two varmints above, you have

S (x-1)/(x+2) dx = S 1 + ( -3 / (x+2) ) dx = x - 3ln|x+2| + C

and

S x/(x+1) dx     = S 1 + ( -1 / (x+1) ) dx = x -  ln|x+1| + C.


David L. Ritter, Ph.D.
Associate Professor
Department of Mathematics
F.I.U.
University Park
Miami, FL 33199