document.write( "Question 1208255: Hi, can you help with this question:
\n" ); document.write( "Use x=alphasintheta for sqrt(alpha^2-x^2), x= alphatantheta for sqrt(alpha^2+x^2) and x= alphasectheta for sqrt(x^2-alpha^2), to find int (x/(sqrt(1-x^2)^3) dx \n" ); document.write( "
Algebra.Com's Answer #846558 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "I don't understand the stuff in the beginning, so I skipped on over\r\n" );
document.write( "to
find int (x/(sqrt(1-x^2)^3) dx
 \r\n" );
document.write( "\r\n" );
document.write( "\"int%28x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29%2Cdx%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"int%28x%5E%22%22%5E%22%22%2F%28%28%281-x%5E2%29%5E%28%221%2F2%22%29%29%5E3%29%2Cdx%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"int%28x%5E%22%22%5E%22%22%2F%28%281-x%5E2%29%5E%28%223%2F2%22%29%29%2Cdx%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"int%28%281-x%5E2%29%5E%28%22-3%2F2%22%29%2Ax%2Adx%29\"\r\n" );
document.write( "\r\n" );
document.write( "Students nowadays do lots more u substituting than they did in \r\n" );
document.write( "the old days when I was teaching.  We taught \"putting in and taking out\",\r\n" );
document.write( "which meant:\r\n" );
document.write( "\r\n" );
document.write( "\"Putting in a factor to cause an expression to become the differential\r\n" );
document.write( "and offsetting by multiplying on the outside of the integral by its reciprocal.\"\r\n" );
document.write( "\r\n" );
document.write( "We would think of, not write, the power rule \"int%28u%5En%2Cdu%29=u%5E%28n%2B1%29%2F%28n%2B1%5E%22%22%29%2BC\",\r\n" );
document.write( "\r\n" );
document.write( "adding 1 to the exponent and dividing by it.\r\n" );
document.write( "\r\n" );
document.write( "You need to have an expression for u, a du, and a constant for an exponent. \r\n" );
document.write( "So for this:\r\n" );
document.write( "\r\n" );
document.write( "\"int%28%281-x%5E2%29%5E%28%22-3%2F2%22%29%2Ax%2Adx%29\"\r\n" );
document.write( "\r\n" );
document.write( "Then we'd think, not write, \"u is (1-x2).\r\n" );
document.write( "Then we'd think, not write, \"we need to get a du\".\r\n" );
document.write( "\r\n" );
document.write( "We would observe that the differential of 1-x2 was -2x*dx, and that\r\n" );
document.write( "we could easily make the x become -2x by multiplying x by -2 and multiplying on\r\n" );
document.write( "the outside of the integral by the reciprocal of -2 which\r\n" );
document.write( "is -1/2.   That would give us our du. So we'd write this:\r\n" );
document.write( "\r\n" );
document.write( "\"expr%28-1%2F2%29int%28%281-x%5E2%29%5E%28%22-3%2F2%22%29%2A%28-2x%29%2Adx%29\" \r\n" );
document.write( "\r\n" );
document.write( "So we'd think, not write, \"u is 1-x2 and the (-2x)dx is du,\r\n" );
document.write( "and then we'd think of the power rule of adding 1 to the exponent and then\r\n" );
document.write( "dividing by it.\r\n" );
document.write( "\r\n" );
document.write( "So we'd add 1 to -3/2 and get -1/2\r\n" );
document.write( "\r\n" );
document.write( "Then we'd divide by the -1/2 and realize that it would cancel with the -1/2\r\n" );
document.write( "we just put on the outside of the integral (to offset the multiplication by -2) and we'd get\r\n" );
document.write( "\r\n" );
document.write( "\"%281-x%5E2%29%5E%28-1%2F2%29%2BC\"  and then\r\n" );
document.write( "\r\n" );
document.write( "\"1%5E%22%22%2Fsqrt%281-x%5E2%29%2BC\"\r\n" );
document.write( "\r\n" );
document.write( "Now if that's too hard for you and you want to do all the substituting that they\r\n" );
document.write( "are requiring today, go to\r\n" );
document.write( "\r\n" );
document.write( "https://www.integral-calculator.com/\r\n" );
document.write( "\r\n" );
document.write( "and they'll go through it step by step for you.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );