Antiderivative Of Secx

Jordan Payton

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08-12-2024

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The antiderivative of sec(x), also known as the indefinite integral of sec(x), isn't immediately obvious. Unlike many trigonometric functions, there's no straightforward formula readily apparent. However, a clever technique involving a specific manipulation allows us to find the solution.

The Method: A Subtle Approach

Finding the antiderivative of sec(x) requires a bit of ingenuity. We don't directly integrate sec(x); instead, we employ a strategy that involves multiplying and dividing by (sec(x) + tan(x)). This might seem arbitrary at first, but observe:

∫sec(x) dx = ∫sec(x) * (sec(x) + tan(x)) / (sec(x) + tan(x)) dx

This seemingly convoluted step is the key. Notice that the derivative of (sec(x) + tan(x)) is (sec(x)tan(x) + sec²(x)). While this doesn't immediately seem helpful, observe how this expression can be factored:

sec(x)tan(x) + sec²(x) = sec(x)(tan(x) + sec(x))

This factored form is crucial. Now, let's rewrite our integral using this information:

∫sec(x) * (sec(x) + tan(x)) / (sec(x) + tan(x)) dx

Let's perform a substitution. Let u = sec(x) + tan(x). Then, du = sec(x)(sec(x) + tan(x)) dx. This elegantly transforms our integral into:

∫1/u du

The Solution: A Logarithmic Result

The integral ∫1/u du is a standard integral that evaluates to ln|u| + C, where C is the constant of integration. Substituting back for u, we obtain the final result:

∫sec(x) dx = ln|sec(x) + tan(x)| + C

Conclusion: A Note on the Constant

Remember the importance of the constant of integration, C. It accounts for the fact that the derivative of a constant is zero. Therefore, any constant added to the antiderivative will still yield sec(x) upon differentiation. This constant is essential for complete accuracy. The antiderivative of sec(x) is not simply ln|sec(x) + tan(x)| but rather ln|sec(x) + tan(x)| + C. This seemingly small detail is significant in calculus.