How do you find the definite integral for: #e^sin(x) * cos(x) dx# for the intervals #[0, pi/4]#?

Question

Charles Arocha · Accepted Answer

Use a #u#-substitution to get #int_0^(pi/4) e^sinx*cosxdx=e^(sqrt(2)/2)-1#.

We'll begin by solving the indefinite integral and then deal with the bounds. In #inte^sinx*cosxdx#, we have #sinx# and its derivative, #cosx#. Therefore we can use a #u#-substitution. Let #u=sinx->(du)/dx=cosx->du=cosxdx#. Making the substitution, we have: #inte^udu# #=e^u# Finally, back substitute #u=sinx# to get the final result: #e^sinx# Now we can evaluate this from #0# to #pi/4#: #[e^sinx]_0^(pi/4)# #=(e^sin(pi/4)-e^0)# #=e^(sqrt(2)/2)-1# #~~1.028#

Adam Adkins · Answer

undefined To find the definite integral of e^sin(x) * cos(x) dx for the interval [0, π/4]:

1. Identify the integral: ∫[0, π/4] e^sin(x) * cos(x) dx.
2. There's no simple antiderivative for e^sin(x) * cos(x), so we'll need to use a numerical method or other techniques to evaluate the integral over this interval.
3. One approach is to use numerical integration methods like Simpson's rule or the trapezoidal rule to approximate the integral.
4. Alternatively, you can use a computer algebra system or integral calculator to compute the definite integral numerically.
   
Therefore, you can approximate the definite integral of e^sin(x) * cos(x) dx for the interval [0, π/4] using numerical integration methods or integral calculators.