document.write( "Question 356418: Please help me solve this equation:
\n" ); document.write( "Evaluate the integral $\"+int+%28+sqrt%28+%281%2Bx%29%2F%281-x%29%29+%29+\"$ \n" ); document.write( "

Algebra.Com's Answer #254485 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"+int+%28+sqrt%28+%281%2Bx%29%2F%281-x%29%29%2C+dx+%29+\"$
\n" ); document.write( "The problem as you posted it is an indefinite integral. So we will not be evaluating (coming up with a numberic value for) it.

\n" ); document.write( "The \"trick\" to a lot of integration problems is to use Algebra (and perhaps Trig) to transform the integral you have into one or more integrals which fit a pattern you know how to identify.
\n" ); document.write( "With our integral, I am going to start by multiplying the numerator and denominator so that the numerator is a perfect square:
\n" ); document.write( " $\"+int+%28+sqrt%28+%28%281%2Bx%29%2F%281-x%29%29%28%281%2Bx%29%2F%281%2Bx%29%29%29%2C+dx+%29+\"$
\n" ); document.write( "This simplifies as follows:
\n" ); document.write( " $\"+int+%28+sqrt%28+%281%2Bx%29%5E2%2F%281-x%5E2%29+%29%2C+dx+%29+\"$
\n" ); document.write( "Using a property of radicals, $\"root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29\"$ , we can split this square root of a fraction into a fraction of square roots:
\n" ); document.write( " $\"+int+%28+sqrt%28+%281%2Bx%29%5E2%29%2Fsqrt%281-x%5E2%29+%2C+dx+%29+\"$
\n" ); document.write( "And the numerator simplifies:
\n" ); document.write( " $\"+int+%28+%281%2Bx%29%2Fsqrt%281-x%5E2%29+%2C+dx+%29+\"$
\n" ); document.write( "(From the original function we know that 1+x must be positive so we do not need to be concerned with +- issues on this square root.)
\n" ); document.write( "We can now split this into separate fractions:
\n" ); document.write( " $\"+int+%28+1%2Fsqrt%281-x%5E2%29++%2B+x%2Fsqrt%281-x%5E2%29%2C+dx+%29\"$
\n" ); document.write( "Now we can use a property of integrals to split this into two:
\n" ); document.write( " $\"+int+%28+1%2Fsqrt%281-x%5E2%29%2C+dx%29++%2B+int+%28+x%2Fsqrt%281-x%5E2%29%2C+dx+%29\"$
\n" ); document.write( "By doing this manipulation of the integral we now have two integrals in a form we should be able to solve. The first one is of the form $\"int+%281%2Fsqrt%28a%5E2+-+u%5E2%29%2C+du%29\"$ with u = x and a = 1. And the second one is of the form of $\"int+%28u%2Fsqrt%28a%5E2-u%5E2%29%2C+du%29\"$ with u = x and a = 1. (In your Calulus book there is probably a table of common forms. You should be able to find both of these in this table so you can see where the solution came from:
\n" ); document.write( "

\r\n" );
document.write( "   -1\r\n" );
document.write( "sin

\n" ); document.write( "Simplifying and combining the two constants of integration into one we get:
\n" ); document.write( "

\r\n" );
document.write( "   -1\r\n" );
document.write( "sin

\n" ); document.write( " \n" ); document.write( "

\n" );