Integrate ? e^x sin e^x dx

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Answer 1 · 2018-04-06T04:56:05

#inte^xsin(e^x)dx=-cos(e^x)+C#

We have #inte^xsin(e^x)dx#.

A substitution will work fine in this case.

#u=e^x#

#(du)/dx=e^x#

#du=e^xdx#

#e^xdx# shows up in the integral, so this substitution is valid. Applying it yields

#intsinudu#

This is an elementary integral.

#intsinudu=-cosu+C#

Recalling that #u=e^x,# we can rewrite in terms of #x.#

#inte^xsin(e^x)dx=-cos(e^x)+C#

Answer 2 · 2018-04-06T15:59:26

this can be done by inspection

#inte^xsine^xdx#

now
#d/(dx)(cosf(x)=-f'(x)sinf(x)#

we have

#d/(dx)(cose^x)=-e^xsine^x#

#:.inte^xsine^xdx=-cose^x+c#