Solving the Integral of Secant (Sec) Using Integration by Parts

integral of sectan

The integral of secant (sec) can be represented as:
∫ sec(x) dx

The integral of secant (sec) can be represented as:
∫ sec(x) dx.

To solve this integral, we can use a technique called integration by parts. This method involves rewriting the integral in terms of two functions, and then applying a formula to evaluate the integral.

Here’s a step-by-step process to solve the integral of secant:

Step 1: Split secant into two parts:
∫ sec(x) dx = ∫ 1 * sec(x) dx.

Step 2: Identify the two functions to integrate by parts:
u = 1 (the function you differentiate first)
dv = sec(x) dx (the function you integrate)

Step 3: Determine the derivatives and antiderivatives:
du = 0 (since the derivative of a constant is zero)
v = ∫ sec(x) dx = ln|sec(x) + tan(x)| (the antiderivative of secant is the natural logarithm of the absolute value of secant plus tangent)

Step 4: Apply the integration by parts formula:
∫ u dv = uv – ∫ v du

Using the formula:
∫ 1 * sec(x) dx = 1 * ln|sec(x) + tan(x)| – ∫ ln|sec(x) + tan(x)| * 0 dx

Simplifying:
∫ sec(x) dx = ln|sec(x) + tan(x)| + C

Therefore, the integral of secant is ln|sec(x) + tan(x)| + C, where C represents the constant of integration.

It’s worth noting that this antiderivative is valid when x ≠ (n * π) + (π/2), where n is any integer. At these specific values, the secant function becomes undefined, and the integral would also be undefined.

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