Integrand Using Series: e^(4t) / (e^(2t) + 3*e^(t) + 2)

[djo201]

The problem is the Integrand of e^(4t)/(e^(2t)+3*e^(t)+2)

Any ideas on how to begin would be greatly appreciated.

 

[Deleted member 4993]

Re: Integrand Using Series

The problem is the Integrand of e^(4t)/(e^(2t)+3*e^(t)+2)

Any ideas on how to begin would be greatly appreciated.

substitute:

u = e^t

 

[fasteddie65]

Just in case you are not sure what to do after the substitution:

u^4/(u^2 + 3u + 2)

Now, use long division to get a more standard integral:

u^2 - 3u + 7 - (15u + 14)/(u^2 + 3u + 2)

Since the denominator is factorable, you can use partial fractions to finish it up.

 

[BigGlenntheHeavy]

\(\displaystyle If \ we \ let \ u = e^{t}, \ then \ du = e^{t}dt \ or \ dt = \frac{du}{u}\)

\(\displaystyle Ergo \ \int{\frac{e^{4t}dt}{e^{2t}+3e^{t}+2}} = \int{\frac{u^{3}du}{u^{2}+3u+2}}, \ not \ \int{\frac{u^{4}du}{u^{2}+3u+2}}.\)

Now use partial fractions and solve.