Evaluate the integral:

the integral of [5e^(2t)]/[1+4e^(2t)]dt.

I used u sub and let u=e^2t and got 5/2arctan(e^2t)+C. But this answer is incorrect. Please help. Thanks

To evaluate the integral ∫(5e^(2t))/(1+4e^(2t)) dt, your initial approach of using a substitution is correct. However, there seems to be an error in your calculations. Let's go through the correct steps together.

Step 1: Let u = e^(2t). Then, differentiate both sides with respect to t to find du/dt:
du/dt = 2e^(2t)

Step 2: Solve the equation from step 1 for dt:
dt = (1/2e^(2t)) du

Step 3: Substitute u and du in the integral:
∫(5e^(2t))/(1+4e^(2t)) dt = ∫(5u)/(1+4u) * (1/2e^(2t)) du
= (5/2) ∫(u/(1+4u)) du

Step 4: Simplify the integrand:
(u/(1+4u)) = (u)/(1+4u) = [1/(4(1+4u)) - 1/4]

Step 5: Substitute the simplified integrand back into the integral:
(5/2) ∫(u/(1+4u)) du = (5/2) ∫[(1/(4(1+4u))) - 1/4] du

Step 6: Evaluate each integral separately:
(5/2) ∫(1/(4(1+4u))) du = (5/2) * (1/4) ln(|1+4u|) + C1

(5/2) ∫(-1/4) du = (-5/2) u + C2

Step 7: Combine the results from step 6:
(5/2) ∫[(1/(4(1+4u))) - 1/4] du = (5/2) [(1/4) ln(|1+4u|) - u] + C

Step 8: Substitute the original u back in terms of t:
(5/2) [(1/4) ln(|1+4u|) - u] + C = (5/2) [(1/4) ln(|1+4e^(2t)|) - e^(2t)] + C

Therefore, the correct solution to the integral is:
(5/2) [(1/4) ln(|1+4e^(2t)|) - e^(2t)] + C.

Please recheck your calculations to see if you made any mistakes along the way.