Math
posted by Dave .
Evaluate the integral of 5^t * sin (5^t) *dt
I started out with u = 5^t , but then I got stuck on du because I am not sure how to take the derivative of 5^t?
The answer from the book is
(1/ln5) cos(5^t) + C
I understand the part with the antiderivative of sin being cos, but that is about it.

You need to recall that
d/dt a^t = ln(a) a^t
because a = e^(lna), so
a^t = e^(lna)^t = e^(lna t)
so, using the chain rule,
d/dt e^(lna t) = lna e^(lna t) = lna a^t
u = 5^t
du = ln5 5^t
∫ 5^t sin(5^t) dt
= ∫1/ln5 sin(u) du
= 1/5 (cos u)
= 1/5 cos(5^t) + C
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