Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(4secx tanx − 2 sec²x)dx

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Recognize that the integral is of the form \(\int (4 \sec x \tan x - 2 \sec^{2} x) \, dx\), which can be split into two separate integrals: \(\int 4 \sec x \tan x \, dx - \int 2 \sec^{2} x \, dx\).

Recall the standard derivatives: \(\frac{d}{dx}(\sec x) = \sec x \tan x\) and \(\frac{d}{dx}(\tan x) = \sec^{2} x\). This helps identify antiderivatives for each term.

For the first integral, \(\int 4 \sec x \tan x \, dx\), use the fact that the derivative of \(\sec x\) is \(\sec x \tan x\), so the antiderivative is \(4 \sec x\) plus a constant.

For the second integral, \(\int 2 \sec^{2} x \, dx\), use the fact that the derivative of \(\tan x\) is \(\sec^{2} x\), so the antiderivative is \(2 \tan x\) plus a constant.

Combine the results from both integrals and add a single constant of integration \(C\) to write the most general antiderivative: \(4 \sec x - 2 \tan x + C\).

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주요 개념

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Indefinite Integral and Antiderivative

An indefinite integral represents the most general form of an antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.

Integration of Trigonometric Functions

Integrating trigonometric functions like secant and tangent requires knowledge of their derivatives and standard integral formulas. Recognizing patterns such as the derivative of sec x being sec x tan x aids in simplifying and solving the integral efficiently.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the solution and helps identify any errors in the integration process, ensuring the integral is accurate.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫(4secx tanx − 2 sec²x)dx