document.write( "Question 564861: Using the method of partial fractions, determine an antiderivative for this function:
\n" ); document.write( "(x^2 + 1) / (x^3 - x^2 - 6x)
\n" ); document.write( "The answer is supposed to be: -1/6 ln x + 2/3 ln | x - 3 | + 1/2 ln | x + 2 | = C
\n" ); document.write( "How to do this? Thank you \n" ); document.write( "
Algebra.Com's Answer #365515 by Earlsdon(6294)\"\" \"About 
You can put this solution on YOUR website!
First decompose the given fraction \"%28x%5E2%2B1%29%2F%28x%5E3-x%5E2-6x%29\" into its partial fractions:
\n" ); document.write( "Factor the denominator:
\n" ); document.write( "\"%28x%5E2%2B1%29%2F%28x%29%28x%2B2%29%28x-3%29\" now we set:
\n" ); document.write( "\"%28x%5E2%2B1%29%2F%28x%5E3-x%5E2-6x%29+=+A%2Fx%2BB%2F%28x%2B2%29%2BC%2F%28x-3%29\" Now add the fractions on the right side.
\n" ); document.write( " Now since the denominators are equal, the numerators must be equal, so we can set:
\n" ); document.write( "\"x%5E2%2B1+=+A%28x%2B2%29%28x-3%29%2BB%28x%29%28x-3%29%2BC%28x%29%28x%2B2%29\" This must be true for all x so it is true for the x-values that make:
\n" ); document.write( "\"x+=+0\"
\n" ); document.write( "\"x%2B2+=+0\" so \"x+=+-2\" and
\n" ); document.write( "\"x-3+=+0\" so \"x+=+3\"
\n" ); document.write( "So we can solve for A, B, and C by letting x = 0, then x = -2, and x = 3.
\n" ); document.write( "After some standard algebra, you'll find that:
\n" ); document.write( "\"A+=+-1%2F6\", \"B+=+-1%2F2\", and \"C+=+2%2F3\" we can now write the partial fraction:
\n" ); document.write( "\"%28x%5E2%2B1%29%2F%28x%5E3-x%5E2-6x%29+=+-1%2F6x-1%2F2%28x%2B2%29%2B2%2F3%28x-3%29\"
\n" ); document.write( "Now when you integrate these partial fractions you'll get the answers:
\n" ); document.write( "\"%28-1%2F6%29lnx-%281%2F2%29ln%28abs%28x%2B2%29%29%2B%282%2F3%29ln%28abs%28x-3%29%29%2BC\" \n" ); document.write( "
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