What is integral of sec x?

Answer: The integral of sec x is ln|sec(x) + tan(x)| + C, where C is the constant of integration.

∫ sec(x) dx = ln ∣sec(x) + tan(x)∣ + C

The integral of sec(x) with respect to x can be found by recognizing its antiderivative. The antiderivative of sec(x) is ln|sec(x) + tan(x)| + C, where ln represents the natural logarithm, sec(x) is the secant function, tan(x) is the tangent function, and C is the constant of integration.

The process involves finding a function whose derivative is sec(x). The derivative of ln|sec(x) + tan(x)| + C with respect to x is sec(x), confirming that it is indeed the antiderivative of sec(x).

The absolute value ensures that the expression remains valid for all x values, and the constant of integration C accounts for the family of functions that differ by a constant. Thus, ln|sec(x) + tan(x)| + C represents the integral of sec(x) with respect to x.

Conclusion:

The integral of sec(x) with respect to x, ∫ sec(x) dx, is represented by the antiderivative ln|sec(x) + tan(x)| + C, where ln denotes the natural logarithm, sec(x) is the secant function, tan(x) is the tangent function, and C is the constant of integration. The process of finding this antiderivative involves recognizing a function whose derivative is sec(x), which leads to the expression ln|sec(x) + tan(x)| + C.

Some Related Questions:

How is the antiderivative of sec(x) derived to be ln|sec(x) + tan(x)| + C?

The antiderivative of sec(x) is determined by recognizing a function whose derivative is sec(x). Through integration techniques, it is found to be ln|sec(x) + tan(x)| + C, where C is the constant of integration.

How does the integral of sec(x) relate to trigonometric identities?

The integral of sec(x) is closely related to trigonometric identities, particularly those involving secant and tangent functions. This integral arises in the evaluation of integrals involving these trigonometric functions and their derivatives.

Read: Integration of Sec x: Formula, Proof and Example