How do you integrate #int e^xsinx# by integration by parts method?
1 Answer
Explanation:
Integration by parts can be expressed:
#int u(x)v'(x) dx = u(x)v(x) - int v(x) u'(x) dx#
Let
Then
and we find:
#int e^x sin x dx = -e^x cos(x) + int e^x cos(x) dx + C_1#
Let
Then
and we find:
#int e^x cos(x) dx = e^x sin(x) - int e^x sin(x) dx + C_2#
Combining these two results, we find:
#int e^x sin x dx = -e^x cos(x) + e^x sin(x) - int e^x sin x dx + (C_1+C_2)#
and hence:
#int e^x sin x dx = 1/2 e^x(sin(x) - cos(x)) + C#
Footnote
Integration by parts is very useful, but can end up leading you down a rabbit hole if you do not choose the parts appropriately.
In the example above, I would instead tend to find the integral by seeing what happens when you differentiate
#d/(dx) e^x sin(x) = e^x sin(x) + e^x cos(x)#
#d/(dx) e^x cos(x) = e^x cos(x) - e^x sin(x)#
So by subtracting the second from the first of these, we find:
#d/(dx) (e^x (sin(x) - cos(x))) = e^x sin(x) + color(red)(cancel(color(black)(e^x(cos(x))))) - color(red)(cancel(color(black)(e^x(cos(x))))) + e^x sin(x)#
#color(white)(d/(dx) (e^x (sin(x) - cos(x)))) = 2e^x sin(x)#
Hence:
#int e^x sin(x) dx = 1/2 e^x (sin(x) - cos(x)) + C#