Evaluating the Integral: ∫ 2sec⁴ x dx

Evaluating integrals is a fundamental skill in calculus, and the integral of 2sec⁴ x dx is a common problem that high school students often encounter. In this article, we will walk through the step-by-step process of solving this integral and discuss the underlying concepts.

Step 1: Identify the Integrand

The integrand in this problem is 2sec⁴ x. This means we need to find the antiderivative (or indefinite integral) of this function.

Step 2: Choose an Appropriate Integration Technique

To solve this integral, we can use the power rule of integration. The power rule states that the integral of x^n dx is (x^(n+1))/(n+1) + C, where C is the constant of integration.

Step 3: Apply the Power Rule

In this case, we have sec⁴ x, which can be rewritten as (sec² x)². Using the power rule, we can integrate as follows:

∫ 2sec⁴ x dx = ∫ 2(sec² x)² dx = 2 ∫ (sec² x)² dx = 2 ∫ sec⁴ x dx = (2/3) ∫ (sec² x)³ dx = (2/3) ∫ (sec² x)² sec² x dx = (2/3) ∫ sec⁶ x dx

Now, we can apply the power rule:

(2/3) ∫ sec⁶ x dx = (2/3) (sec⁴ x)/4 + C = (1/6) sec⁴ x + C

Step 4: Verify the Solution

To verify the solution, we can differentiate the antiderivative to ensure that we get the original integrand:

d/dx (1/6 sec⁴ x + C) = (1/6) 4 sec⁴ x tan x = 2 sec⁴ x

The differentiation result matches the original integrand, confirming that the solution is correct.

Conclusion

In this article, we have demonstrated the step-by-step process of evaluating the integral of 2sec⁴ x dx. By applying the power rule of integration, we were able to find the antiderivative and express the final solution in a concise form. Understanding the techniques and strategies used in this problem is crucial for success in high school calculus.