Question 216792
{{{int(e^(-sin(x))*cos(x)*dx)}}}
Since we are given the substitution to use the solution is pretty straightforward. Start with:
u = sin(x)
Express the derivative of each side:
du/dx = cos(x)
Multiply both sides by dx:
du = cos(x)*dx
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:
{{{int(e^(-u)*du)}}}
In this form the integral is easy to find:
{{{int(e^(-u)*du)}}} = {{{-e^(-u) + C}}}
And finally we substitute back sin(x) for u:
{{{int(e^(-u)*du)}}} = {{{-e^(-u) + C = -e^(-sin(x)) + C }}}