Find the indefinite integral of sin(x)*e^x
As we are integrating, we must decide which method to use. As the integrand is of the form f(x)*g(x), integration by parts seems to make sense. Firstly, let L = INT(sin(x)*e^x). So we want to find L - this will help later.
Let u = sin(x), so du/dx = cos(x), and let dv/dx = e^x, so v = e^x.
Therefore L = sin(x)*e^x - INT(cos(x)*e^x)) (this formula is given to us).
To deal with the second term, we use integration by parts again.
Let u = cos(x), so du/dx = -sin(x), and let dv/dx = e^x, so v = e^x.
Therefore L = sin(x)*e^x - [cos(x)*e^x + INT(sin(x)*e^x)] - but this last term is simply L, our original expression!
Rearranging yields 2L = sin(x)*e^x - cos(x)*e^x.
Now simply divide by 2 and factorise the e^x, giving the final answer of:
L = e^x(sin(x) - cos(x))/2 + c
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