23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (sec² Θ + sec Θ tan Θ)dΘ

Verified step by step guidance

1. Rewrite the integral as the sum of two separate integrals: ∫(sec²Θ)dΘ + ∫(secΘ tanΘ)dΘ. This is possible because the integral of a sum is the sum of the integrals.
2. Recognize the standard integral formulas: ∫(sec²Θ)dΘ = tanΘ + C₁ and ∫(secΘ tanΘ)dΘ = secΘ + C₂, where C₁ and C₂ are constants of integration.
3. Combine the results of the two integrals: tanΘ + secΘ + C, where C is the combined constant of integration (C = C₁ + C₂).
4. Verify your solution by differentiating the result, tanΘ + secΘ + C. Use the derivative rules: d/dΘ(tanΘ) = sec²Θ and d/dΘ(secΘ) = secΘ tanΘ.
5. Check that the derivative of tanΘ + secΘ + C matches the original integrand, sec²Θ + secΘ tanΘ. If it does, your solution is correct.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation.

Trigonometric Functions

Trigonometric functions, such as secant (sec) and tangent (tan), are fundamental in calculus, particularly in integration and differentiation. The secant function is defined as the reciprocal of the cosine function, while the tangent function is the ratio of sine to cosine. Understanding their derivatives and integrals is crucial for solving problems involving these functions.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values. This theorem is essential for verifying the correctness of indefinite integrals by differentiating the result to check if it matches the original integrand.