Simplifying and Solving the Integral of Sec(x) | Step-by-Step Guide

integral of secx

The integral of sec(x) can be a bit tricky to evaluate directly

The integral of sec(x) can be a bit tricky to evaluate directly. However, it can be simplified and solved using a technique called substitution. Here’s the step-by-step process to find the integral of sec(x):

Step 1: Write the integral as ∫sec(x) dx.

Step 2: Multiply the numerator and the denominator of the integrand by (sec(x) + tan(x)). This is done to convert sec(x) into (sec(x) + tan(x)) which simplifies the integration process.

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

Step 3: Simplify the integrand:

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

Step 4: Apply substitution. Let u = sec(x) + tan(x), and du = (sec(x)tan(x) + sec^2(x)) dx. This allows us to rewrite the integral in terms of u:

∫(sec(x) + tan(x))/sec(x) dx = ∫du

Step 5: Integrate the right side of the equation, which gives us:

∫du = u + C

Step 6: Substitute back u = sec(x) + tan(x) into the result:

∫sec(x) dx = sec(x) + tan(x) + C

So, the integral of sec(x) is sec(x) + tan(x) + C, where C is the constant of integration.

It’s worth noting that an alternative way to approach this integral is by using the natural logarithm function. The integral of sec(x) can be expressed as ln|sec(x) + tan(x)| + C. Both answers are correct, as they differ only by a constant.

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