document.write( "Question 445411: Find the integral\r
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Algebra.Com's Answer #306829 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"int%28+%28x%2Asqrt%281-x%5E2%29+dx%29%29\"\r\n" );
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document.write( "Rewrite as\r\n" );
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document.write( "\"int%28%28++%281-x%5E2%29%5E%281%2F2%29x%2Adx++%29%29\"\r\n" );
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document.write( "Let \"u=1-x%5E2\"\r\n" );
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document.write( "Then \"du=-2x%2Adx\"\r\n" );
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document.write( "Solve for dx.  \"dx=%28du%29%2F%28-2x%29\"\r\n" );
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document.write( "Substitute for u and du\r\n" );
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document.write( "\"int%28%28++%281-x%5E2%29%5E%281%2F2%29x%2Adx++%29%29\"\r\n" );
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document.write( "\"int%28%28++u%5E%281%2F2%29x%2Aexpr%28du%2F%28-2x%29%29++%29%29\"\r\n" );
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document.write( "Cancel the x's\r\n" );
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document.write( "\"int%28%28++u%5E%281%2F2%29cross%28x%29%2Aexpr%28du%2F%28-2cross%28x%29%29%29++%29%29\"\r\n" );
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document.write( "The -2 on the bottom is the same as a multiplication by\r\n" );
document.write( "\"-1%2F2\" and can be brought out in front of the integral:\r\n" );
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document.write( "\"expr%28-1%2F2%29int%28%28++u%5E%281%2F2%29du%29++%29%29\"\r\n" );
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document.write( "Next we use the formula \"int%28u%5En%2Adu%29\"\"%22%22=%22%22\"\"expr%28u%5E%28n%2B1%29%2F%28n%2B1%29%29%2BC\"\r\n" );
document.write( "with \"n=1%2F2\"\r\n" );
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document.write( "\"expr%28-1%2F2%29int%28u%5E%281%2F2%29%2Adu%29\"\"%22%22=%22%22\"\"expr%28-1%2F2%29%28u%5E%281%2F2%2B1%29%2F%281%2F2%2B1%29%29%2BC\"\"%22%22=%22%22\"\"expr%28-1%2F2%29%28u%5E%283%2F2%29%2F%283%2F2%29%29%2BC\"\r\n" );
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document.write( "The \"3%2F2\" on the bottom is a division by \"3%2F2\" which is the\r\n" );
document.write( "same as a multiplication by \"2%2F3\" which we can take out in front\r\n" );
document.write( "of the integral sign.  So we have:\r\n" );
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document.write( "\"expr%282%2F3%29%2Aexpr%28-1%2F2%29%28u%5E%283%2F2%29%29%2BC\"\"%22%22=%22%22\"\"expr%28-1%2F3%29%28u%5E%283%2F2%29%29%2BC\"\r\n" );
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document.write( "Finally we replace u using \"u=1-x%5E2\"\r\n" );
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document.write( "\"expr%28-1%2F3%29%281-x%5E2%29%5E%283%2F2%29%2BC\"\r\n" );
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document.write( "That's a long process, but later you will learn this shortcut:\r\n" );
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document.write( "\"int%28%28++%281-x%5E2%29%5E%281%2F2%29x%2Adx++%29%29\"\r\n" );
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document.write( "Since the derivative of \"1-x%5E2\" is \"-2x\", and we\r\n" );
document.write( "already have an x factor, we only need to put in \"-2\"\r\n" );
document.write( "and take out \"-1%2F2\", since they would cancel:\r\n" );
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document.write( "\"red%28expr%28-1%2F2%29%29int%28%28++%281-x%5E2%29%5E%281%2F2%29red%28-2%29xdx++%29%29\"\r\n" );
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document.write( "Then you just observe that that is in the form \"int%28u%5En%2Adu%29\"\r\n" );
document.write( "without writing any u's or du's.\r\n" );
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document.write( "Then you add 1 to the exponent and multiply by the reciprocal\r\n" );
document.write( "of the new exponent:\r\n" );
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document.write( "\"expr%28-1%2F2%29expr%282%2F3%29%281-x%5E2%29%5E%283%2F2%29%2BC\"\r\n" );
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document.write( "\"expr%28-1%2F3%29%281-x%5E2%29%5E%283%2F2%29%2BC\"\r\n" );
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document.write( "Edwin

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