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 given integral ∫(5e^(2t))/(1+4e^(2t)) dt, you correctly used the substitution u = e^(2t). However, it seems like there was a mistake in the integration step using this substitution.
Let's go through the correct steps to evaluate this integral using u-substitution:
Step 1: Determine the new expression for dt in terms of du using the substitution u = e^(2t):
Differentiate both sides of u = e^(2t) with respect to t:
du/dt = 2e^(2t)
Rearrange the expression to solve for dt:
dt = (1/2e^(2t)) du
Step 2: Rewrite the integral in terms of u:
Now substitute u = e^(2t) and dt = (1/2e^(2t))du into the original integral:
∫(5e^(2t))/(1+4e^(2t)) dt = ∫(5/2) (du)/(1+4u)
Step 3: Simplify the integrand:
The expression (5/2)/(1+4u) can be split into partial fractions:
(5/2)/(1+4u) = A/(1+4u), where A is a constant.
Multiply both sides of the equation by (1+4u) to eliminate the denominator:
(5/2) = A
Thus, A = 5/2.
So, the integrand can be written as:
(5/2) / (1+4u) = (5/2) / (1+4e^(2t))
Step 4: Evaluate the integral:
Now we can integrate the simpler expression:
∫(5/2) / (1+4u) du = (5/2) ln|1+4u| + C
Step 5: Substitute back:
Remember that u = e^(2t). Substituting this back into the equation:
(5/2) ln|1+4u| + C = (5/2) ln|1+4e^(2t)| + C
Therefore, the correct answer is (5/2) ln|1+4e^(2t)| + C, not 5/2arctan(e^(2t)) + C.