Hello, I am working through CT1 and have come across the worked example whereby the integral of (0.5t+2)*exp(-(0.005t^2+0.04t)) dt becomes the integral of -50 e^u du via the use of substitution, I understand/can utilise the concept of substitution and how it has been applied here however should there be a case whereby substitution is not applicable (0.5t+2 is not a multiple of u') can someone provide a step by step method of integrating this example please as I am struggling to work this through without substitution? Alternatively, if anyone on this forum could confirm that should a question similar to the above come up in the exam that substitution as per the example would be sufficient, this would be appreciated. Many Thanks Richard
It's not easy without substitution mainly because of \(t^2\) term in the power of \(e\) You won't see any intergral in the exam which'll require more than substitution or integration by parts. Alternatively, you can purchase "TI30X Pro" calculator which can do integration.
I suppose we could get integration by parts - eg \(te^t\) but this hasn't been asked (though it is in the core reading in the derivation of one of the continuous increasing annuities). Otherwise integration would require complex analysis which is way beyond the actuarial exams.
