document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=229536>Question 229536</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate the following indefinite integral:<BR>\n" );
document.write( "(5)/(x^(2)-x-6) dx (use partial fractions) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #170163 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=170163\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To integrate by partial fractions, you need to factor the denominator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%285%2F%28x%5E2-x-6%29%2C+dx%29+=+int%285%2F%28%28x-3%29%28x%2B2%29%29%2C+dx%29 ALIGN=MIDDLE  ALT=\"int%285%2F%28x%5E2-x-6%29%2C+dx%29+=+int%285%2F%28%28x-3%29%28x%2B2%29%29%2C+dx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we want to rewrite the integrand as a sum of fractions where the denominators of the fractions are the factors of the current denominator. (This is a little more complicated if you have an exponent with a exponent of 2 or more. Fortunately we are not faced with this.) So we want to rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F%28%28x-3%29%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"5%2F%28%28x-3%29%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2F%28x-3%29+%2B+B%2F%28x%2B2%29 ALIGN=MIDDLE  ALT=\"A%2F%28x-3%29+%2B+B%2F%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > We just have to figure out what values for A and B make this work. To figure this out we need to think, how would we add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2F%28x-3%29+%2B+B%2F%28x%2B2%29 ALIGN=MIDDLE  ALT=\"A%2F%28x-3%29+%2B+B%2F%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >? Answer: Get the denominators the same and then add. Now we have to ask: How do we get the denominators the same? Answer: Multiply the numerator and denominator of each fraction by the other fraction's denominator. In our case:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2F%28x-3%29+%2B+B%2F%28x%2B2%29+=+%28A%2F%28x-3%29%29%28%28x%2B2%29%2F%28x%2B2%29%29+%2B+%28B%2F%28x%2B2%29%29%28%28x-3%29%2F%28x-3%29%29+=+%28A%28x%2B2%29%29%2F%28%28x-3%29%28x%2B2%29%29+%2B+%28B%28x-3%29%29%2F%28%28x-3%29%28x%2B2%29%29+=+%28A%28x%2B2%29+%2B+B%28x-3%29%29%2F%28%28x-3%29%28x%2B2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And we want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%28x%2B2%29+%2B+B%28x-3%29%29%2F%28%28x-3%29%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"%28A%28x%2B2%29+%2B+B%28x-3%29%29%2F%28%28x-3%29%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F%28%28x-3%29%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"5%2F%28%28x-3%29%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will be true if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%28x%2B2%29+%2B+B%28x-3%29%29+=+5 ALIGN=MIDDLE  ALT=\"%28A%28x%2B2%29+%2B+B%28x-3%29%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. (If you prefer formulas instead of all this logic, here is one that will work for a fraction with two factors in the denominator with no factor having an exponent of 2 or more:<BR>\n" );
document.write( "(A*(B's denominator) + B*(A's denominator) = (current numerator)<br><BR>\n" );
document.write( "We can solve this equation. Simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax+%2B+2A+%2B+Bx+-3B+=+5 ALIGN=MIDDLE  ALT=\"Ax+%2B+2A+%2B+Bx+-3B+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Grouping the variable and constant terms and factoring:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28Ax+%2B+Bx%29+%2B+%282A+%2B+%28-3B%29%29+=+5 ALIGN=MIDDLE  ALT=\"%28Ax+%2B+Bx%29+%2B+%282A+%2B+%28-3B%29%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A+%2B+B%29x+%2B+%282A+%2B+%28-3B%29%29+=+5 ALIGN=MIDDLE  ALT=\"%28A+%2B+B%29x+%2B+%282A+%2B+%28-3B%29%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the right side has no x term, the left side must have one either. This will be true if<BR>\n" );
document.write( "A + B = 0<BR>\n" );
document.write( "The right side has a constant term of 5 so the left side must have a constant term of 5:<BR>\n" );
document.write( "2A + (-3B) = 5<BR>\n" );
document.write( "We can now solve this system of A and B. Solving the first equation for B we get:<BR>\n" );
document.write( "B = -A<BR>\n" );
document.write( "Substituting this into the other equation we get:<BR>\n" );
document.write( "2A + (-3(-A)) = 5<BR>\n" );
document.write( "2A + 3A = 5<BR>\n" );
document.write( "5A = 5<BR>\n" );
document.write( "A = 1<BR>\n" );
document.write( "B = -A = -(1)<BR>\n" );
document.write( "Finally we have our partial fractions!<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%285%2F%28%28x-3%29%28x%2B2%29%29%2C+dx%29+=+int%281%2F%28x-3%29+%2B+%28-1%29%2F%28x%2B2%29%2C+dx%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The new integrals are relatively easy. They are ln's:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%285%2F%28%28x-3%29%28x%2B2%29%29%2C+dx%29+=+int%281%2F%28x-3%29+%2B+%28-1%29%2F%28x%2B2%29%2C+dx%29+=+ln%28abs%28x-3%29%29+-+ln%28abs%28x%2B2%29%29+%2B+C ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using a properties of logarithms and absolute values, we can combine the the two ln's:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%285%2F%28%28x-3%29%28x%2B2%29%29%2C+dx%29+=+int%281%2F%28x-3%29+%2B+%28-1%29%2F%28x%2B2%29%2C+dx%29+=+ln%28abs%28x-3%29%29+-+ln%28abs%28x%2B2%29%29+%2B+C+=+ln%28abs%28%28x-3%29%2F%28x%2B2%29%29%29+%2B+C ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );