document.write( "Question 1193106: What is the integral of dx/(e^(2x)+1) using trigonometric substitution? \n" ); document.write( "
Algebra.Com's Answer #825198 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"int%28%28dx%29%2F%28e%5E%282x%29%2B1%29%29\"\r\n" );
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document.write( "Notice that \"e%5E%282x%29%2B1\" can be written as the sum of squares:\r\n" );
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document.write( "\"%28e%5Ex%29%5E2%2B1%5E2\" \r\n" );
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document.write( "Draw a right triangle\r\n" );
document.write( "Always put the simpler expression on the adjacent side.  Since 1\r\n" );
document.write( "is simpler than ex, we put it on the adjacent side.  \r\n" );
document.write( "The sign between the squares is + so we put the more complicated \r\n" );
document.write( "expression on the opposite side.  [If it had been - we would have \r\n" );
document.write( "put the more complicated expression on the hypotenuse.]\r\n" );
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document.write( "So \"e%5Ex%2F1%5E%22%22+=+opposite%2Fadjacent+=+tan%28theta%29\"\r\n" );
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document.write( "\"e%5Ex+=+tan%28theta%29\"\r\n" );
document.write( "Solve for x by taking natural logs of both sides:\r\n" );
document.write( "\"x=ln%28tan%28theta%29%5E%22%22%29\"\r\n" );
document.write( "We find dx by multiplying the derivative by dθ \r\n" );
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document.write( "\"dx=sec%5E2%28theta%29%2Amatrix%281%2C2%2Cd%2Ctheta%29%2Ftan%28theta%29%29\"\r\n" );
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document.write( "Then we use the Pythagorean theorem to label the\r\n" );
document.write( "missing side, which is the hypotenuse.\r\n" );
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document.write( "Now we're ready to substitute:\r\n" );
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document.write( "\"int%28%28dx%29%2F%28e%5E%282x%29%2B1%29%29\"\r\n" );
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document.write( "We substitute for dx\r\n" );
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document.write( "Now here's a trick.  Observe that the denominator \r\n" );
document.write( "equals the square of the hypotenuse, and since \"sec%28theta%29=hypotenuse%2Fadjacent=sqrt%28%28e%5E%282x%29%2B1%29%29%2F1%5E%22%22\"\r\n" );
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document.write( "the denominator is just sec2(θ)\r\n" );
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document.write( "Now the sec2(θ)'s cancel:\r\n" );
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document.write( "\"int%28matrix%281%2C2%2Cd%2Ctheta%29%2Ftan%28theta%29%29\"\r\n" );
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document.write( "\"int%28cot%28theta%29matrix%281%2C2%2Cd%2Ctheta%29%29\"\r\n" );
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document.write( "\"ln%28sin%28theta%29%5E%22%22%29%2BC\"\r\n" );
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document.write( "Now we look back at the triangle and find that\r\n" );
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document.write( "\"sin%28theta%29=opposite%2Fhypotenuse+=+e%5Ex%2Fsqrt%28e%5E%282x%29%2B1%29\"\r\n" );
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document.write( "\"ln%28e%5Ex%2Fsqrt%28e%5E%282x%29%2B1%29%29%2BC\"\r\n" );
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document.write( "\"ln%28e%5Ex%29-ln%28sqrt%28e%5E%282x%29%2B1%29%29%2BC\"\r\n" );
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document.write( "\"x-ln%28e%5E%282x%29%2B1%29%5E%28%221%2F2%22%29%2BC\"\r\n" );
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document.write( "\"x-expr%281%2F2%29ln%28e%5E%282x%29%2B1%29%2BC\"   <-- final answer\r\n" );
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document.write( "Edwin
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