Question

Hi, could someone please help me with this hw question asap?
Given that the integral of (e^x*sin(5x))dx=((e^x)/26)*(sin(5x)-5cos(5x))+c, evaluate the integral from 1 to e^(pi/10) of sin(5ln(x))dx

Answer 1

kind of tricky, eh? Let x = e^u. Then

u = lnx
du = 1/x dx, so
dx = x du = e^u du

Now the integral is

∫ sin(5lnx) dx
= ∫ sin(5u) e^u du
= ∫ e^u sin(5u) du

and you can apply your formula.