This is a differential equation problem, but since I have problem with integration, I am posting here for help. Thank you very much.

I need to integrate the right side

I will list my steps.

1. perform integration by part

2. result

3. again, integration by part

4. result (edited)

The last result is not right. This is the result from the website (Paul's Note)

Pauls Online Notes : Differential Equations - Modeling with First Order DE's (first problem)

I don't even have (200+t)^3

Can someone help me on this? Thank you very much!