Use the substitution formula to evaluate the integral.

sec2 2x dx

Question

Use the substitution formula to evaluate the integral.

sec2 2x dx

BlackTom AI · Accepted Answer

To tackle the integral, I’ll consider a substitution that matches the structure: the derivative of tan(2x) involves sec^2(2x).

Option by option reasoning:

Option 1: dash and -e/2. This odd entry seems to try to place a negative half of e, which would arise if the substitution produced a negative sign or if the integral evaluated to -(1/2) e. However, the substitution u = tan(2x) yields du = 2 sec^2(2x) dx, so sec^2(2x) dx becomes du/2, which is positive. So a negative factor here does not fit the algebra.

Option 2: e/2 (the positive half of e). This matches the result you get after performing the substitution correctly. With u = tan(2x), du = 2 sec^2(2x) dx, hence sec^2(2x) dx = du/2. The integral becomes (1/2) ∫_{u=0}^{1} (1 + e^u) du, which evaluates to (1/2) [u + e^u]_{0}^{1} = (1/2) [(1 + e) - (0 + 1)] = (1/2) e. The steps align, and this option corresponds to the proper evaluation.

Option 3: e. This would imply the integral equals e exactly, missing the factor of 1/2 and the contribution from the lower limit, so it’s inconsistent with the computed expression (1/2) [u + e^u] from 0 to 1. It disregards both the 1 and the exponential term’s integration behavior over the bounds.

Option 4: 2e. This would arise if the integral contributed twice the e-term or if the substitution yielded a factor of 4 somewhere, which is not the case here. The 2 in the substitution only cancels with the 1/2 outside, not producing a 2e result; the algebra does not support a 2e magnitude.

Breaking down the core steps: set u = tan(2x). Then du = 2 sec^2(2x) dx, so sec^2(2x) dx = du/2. The limits change from x = 0 to x = pi/8 into u = tan(0) = 0 and u = tan(pi/4) = 1. The integral becomes (1/2) ∫_{0}^{1} (1 + e^u) du. Integrating, we get (1/2) [u + e^u]_{0}^{1} = (1/2) e. The evaluation is consistent with option 2 and inconsistent with the other options.

Use the substitution formula to evaluate the integral. sec2 2x dx Single choice

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