SOLUTION: Hi all, can somebody please help me with the following? I have to evaluate the following intergral using the given substitution. ∫e^-(sin x)cos x dx, substitution u = sinx

Algebra ->  Equations -> SOLUTION: Hi all, can somebody please help me with the following? I have to evaluate the following intergral using the given substitution. ∫e^-(sin x)cos x dx, substitution u = sinx       Log On


   

Question 216792: Hi all, can somebody please help me with the following? I have to evaluate the following intergral using the given substitution.
∫e^-(sin x)cos x dx, substitution u = sinx
Any help would be great.
Thanks, -Nick.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
int%28e%5E%28-sin%28x%29%29%2Acos%28x%29%2Adx%29
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%28e%5E%28-u%29%2Adu%29
In this form the integral is easy to find:
int%28e%5E%28-u%29%2Adu%29 = -e%5E%28-u%29+%2B+C
And finally we substitute back sin(x) for u:
int%28e%5E%28-u%29%2Adu%29 = -e%5E%28-u%29+%2B+C+=+-e%5E%28-sin%28x%29%29+%2B+C+