So for problem 104 in the link below, I have a continuous perpetuity with the following properties:

payment of 1 for 0<t<10

payment of 1.03^t-10 for t>=10

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i is annual effective rate of .06 and they want the PV at t =0. I can solve the PV when the payment is 1 and get that portion of the printed solution but the part with the increasing payment I cannot seem to get to the integral in the printed solution.

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I start with discounting 10 years so 1.06^-10 and the integral from 10 to infinity of the function 1.06^-t *1.03^t-10. Then I move the -10 outside to get (1.06*1.03)^-10 times the integral from 10 to infinity of 1.06^-t *1.03^t which I work out as (1.06*1.03)^-10 * (1.03/1.06)^t * ln(1.03/1.06) evaluated from 0 to infinity.

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Which I then solve as 10.86. The solution says this portion should evaluate to 19.45

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I know the solution has some sort of substitution going on but I cannot wrap my hear around how they got to the solution. Perhaps I am missing something that looks simple but if anyone can help, that would be great.

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Problem sets: https://www.soa.org/globalassets/assets/Files/Edu/2018/2018-10-exam-fm-sample-questions.pdf

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Solutions:https://www.soa.org/globalassets/assets/Files/Edu/2018/2018-10-exam-fm-sample-solutions.pdf