Question

Integrate ∫tan(2x)sec^2(2x) with respect to x using the substitution method, where U = sec(2x).

Answer 1

To integrate tan(2x)sec^2(2x), use the substitution U = sec(2x), resulting in the antiderivative ¼ sec^2(2x) + C.

Step-by-step explanation:

To integrate the function ∫tan(2x)sec²(2x) with respect to x using the substitution method, we can let U = sec(2x). This substitution simplifies the integration process as follows:

1. First, observe that the derivative of sec(2x) is 2sec(2x)tan(2x), which is closely related to our integrand.
2. Next, differentiate U to get dU = 2sec(2x)tan(2x) dx, or dx = dU / (2tan(2x)sec(2x)).
3. Substitute sec(2x) for U and dx from the previous step into our integral, transforming ∫ tan(2x)sec²(2x) dx into ½ ∫ U dU.
4. The resulting integral is ½ U² / 2 + C, where C is the constant of integration.
5. Finally, substitute back in sec(2x) for U to find the antiderivative in terms of x: ½ sec²(2x) / 4 + C, or ¼ sec²(2x) + C.

This example demonstrates the power of substitution in simplifying integral calculus problems.