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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.

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  1. 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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