document.write( "Question 216792: Hi all, can somebody please help me with the following? I have to evaluate the following intergral using the given substitution.
\n" ); document.write( "∫e^-(sin x)cos x dx, substitution u = sinx
\n" ); document.write( "Any help would be great.
\n" ); document.write( "Thanks, -Nick. \n" ); document.write( "

Algebra.Com's Answer #163738 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"int%28e%5E%28-sin%28x%29%29%2Acos%28x%29%2Adx%29\"$
\n" ); document.write( "Since we are given the substitution to use the solution is pretty straightforward. Start with:
\n" ); document.write( "u = sin(x)
\n" ); document.write( "Express the derivative of each side:
\n" ); document.write( "du/dx = cos(x)
\n" ); document.write( "Multiply both sides by dx:
\n" ); document.write( "du = cos(x)*dx
\n" ); document.write( "We can see the right side of the last equation in our integral. We can also see the right side of u = sin(x) in the integral. So we can substitute for both of these giving:
\n" ); document.write( " $\"int%28e%5E%28-u%29%2Adu%29\"$
\n" ); document.write( "In this form the integral is easy to find:
\n" ); document.write( " $\"int%28e%5E%28-u%29%2Adu%29\"$ = $\"-e%5E%28-u%29+%2B+C\"$
\n" ); document.write( "And finally we substitute back sin(x) for u:
\n" ); document.write( " $\"int%28e%5E%28-u%29%2Adu%29\"$ = $\"-e%5E%28-u%29+%2B+C+=+-e%5E%28-sin%28x%29%29+%2B+C+\"$
\n" ); document.write( " \n" ); document.write( "

\n" );