1. Simple Integration

I have an integrating factor of $e^(3lnt)$and have simplified it to $e^t^3$ near the end of the problem we are required to integrate $-3t^3 dt$ which comes to $-3/4t^4$ right?

Now my friend has kept the integrating factor in the form $e^(3lnt)$ throughout the problem and thus has to integrate $-3e^(3lnt) dt$ at the end of the problem. She, however, thinks that this becomes $-te^(3lnt)$.

Obviosuly one of us is wrong; but who?

2. Originally Posted by AmberLamps
I have an integrating factor of $e^(3lnt)$and have simplified it to $e^{t^3}$ near the end of the problem we are required to integrate $-3t^3 dt$ which comes to $-3/4t^4$ right?

Now my friend has kept the integrating factor in the form $e^(3lnt)$ throughout the problem and thus has to integrate $-3e^(3lnt) dt$ at the end of the problem. She, however, thinks that this becomes $-te^(3lnt)$.

Obviosuly one of us is wrong; but who?
If you post the problem, I could tell you for sure, but given an integrating factor of $e^{3ln(t)}$, I would simplify this to $e^{ln(t^3)}$. The log and the exponential then cancel out to give you an integrating factor of $t^3$. Is this the integrating factor that you actually got?

Secondly, how did your friend get $-te^{3ln(t)}$? Anycase, he is wrong, if you take the super complicated and long way around, you still get the integration of $\frac{e^{4ln(t)}}{4}=\frac{t^4}{4}$.

Out of interest, if you want to do the long way, let u = ln(t), hence t = exp(u) then let v =4u. Don't for get to change the dt in each case.