How do you find the integral of # xe^x - sec(7x)tan(7x) dx#?

1 Answer
Apr 21, 2018

#int(xe^x-sec7xtan7x)dx=xe^x-e^x-1/7sec(7x)+C#

Explanation:

So, we want

#int(xe^x-sec7xtan7x)dx#. We can split up across the difference, yielding the following two integrals:

#intxe^xdx-intsec7xtan7xdx#

For #intxe^xdx#, we will use Integration by Parts, making the following selections:

#u=x#
#du=dx#
#dv=e^xdx#
#v=inte^xdx=e^x#

#uv-intvdu=xe^x-inte^xdx#

#=xe^x-e^x#

For #intsec7xtan7xdx#, let's first make a simple substitution to clean things up:

#u=7x#
#du=7dx#
#1/7du=dx#

Then, we have the common integral

#1/7intsecutanudu=1/7secu=1/7sec(7x)#

Combining our integrals together and putting in the constant of integration, we get

#int(xe^x-sec7xtan7x)dx=xe^x-e^x-1/7sec(7x)+C#